🪷On Mind: On Reasoning (1990-1999)

"It is difficult to distinguish deduction from what in other circumstances is called problem-solving. And concept learning, inference, and reasoning by analogy are all instances of inductive reasoning." (Detectives typically induce, rather than deduce.) None of these things can be done separately from each other, or from anything else. They are pseudo-categories." (Frank Smith, "To Think: In Language, Learning and Education", 1990)

"Common sense is merely unaided intuition, and unaided intuition is reasoning performed in the absence of instruments and the tested knowledge of science." (Edward O Wilson, "The Diversity of Life", 1992)

"All this points to an appealing intuition: that a faculty for analogical reasoning is an innate part of human cognition." (Dedre Gentner & Michael Jeziorski,"Western science", 1993)

"The deep paradox uncovered by AI research: the only way to deal efficiently with very complex problems is to move away from pure logic. [...] Most of the time, reaching the right decision requires little reasoning.[...] Expert systems are, thus, not about reasoning: they are about knowing. [...] Reasoning takes time, so we try to do it as seldom as possible. Instead we store the results of our reasoning for later reference." (Daniel Crevier, "The Tree of Knowledge", 1993)

"The pinball machine is one of those rare dynamical systems whose chaotic nature we can deduce by pure qualitative reasoning, with fair confidence that we have not wandered astray. Nevertheless, the angles in the paths of the balls that are introduced whenever a ball strikes a pin and rebounds […] render the system some what inconvenient for detailed quantitative study." (Edward N Lorenz, "The Essence of Chaos", 1993)

“Mathematicians apparently don’t generally rely on the formal rules of deduction as they are thinking. Rather, they hold a fair bit of logical structure of a proof in their heads, breaking proofs into intermediate results so that they don’t have to hold too much logic at once. In fact, it is common for excellent mathematicians not even to know the standard formal usage of quantifiers" (for all and there exists), yet all mathematicians certainly perform the reasoning that they encode." (William P Thurston,"On Proof and Progress in Mathematics", 1994)

“Mathematicians apparently don’t generally rely on the formal rules of deduction as they are thinking. Rather, they hold a fair bit of logical structure of a proof in their heads, breaking proofs into intermediate results so that they don’t have to hold too much logic at once. In fact, it is common for excellent mathematicians not even to know the standard formal usage of quantifiers" (for all and there exists), yet all mathematicians certainly perform the reasoning that they encode." (William P Thurston,"On proof and progress in mathematics", Bulletin of the American Mathematical Society Vol. 30" (2), 1994)

“Mathematics is not the study of an ideal, preexisting nontemporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather, it is the part of human studies which is capable of achieving a science-like consensus, capable of establishing reproducible results. The existence of the subject called mathematics is a fact, not a question. This fact means no more and no less than the existence of modes of reasoning and argument about ideas which are compelling an conclusive, ‘noncontroversial when once understood’." (Philip J Davis & Rueben Hersh,"The Mathematical Experience", 1995)

"Scientists reach their  conclusions  for the damnedest of reasons: intuition, guesses, redirections after wild-goose chases, all combing with a dollop of rigorous observation and logical  reasoning to be sure […] This  messy and personal side of science should not be  disparaged, or covered up, by  scientists for two  major reasons. First, scientists should proudly show this  human face to  display their kinship with all other  modes of creative human thought […] Second, while biases and references often impede understanding, these  mental idiosyncrasies  may  also serve as powerful, if  quirky and personal, guides to solutions." (Stephen J Gould, "Dinosaur in a  Haystack: Reflections in natural  history", 1995)

"Scientists reach their  conclusions  for the damnedest of reasons: intuition, guesses, redirections after wild-goose chases, all combing with a dollop of rigorous observation and logical  reasoning to be sure […]" (Stephen J Gould, "Dinosaur in a  Haystack: Reflections in natural  history", 1995)

"To select an appropriate fuzzy implication for approximate reasoning under each particular situation is a difficult problem. Although some theoretically supported guidelines are now available for some situations, we are still far from a general solution to this problem." (George Klir, "Fuzzy sets and fuzzy logic", 1995)

“Suppose that we think of the integers lined up like dominoes. The inductive step tells us that they are close enough for each domino to knock over the next one, the base case tells us that the first domino falls over; the conclusion is that they all fall over. The fault in this analogy is that it takes time for each domino to fall and so a domino which is a long way along the line won't fall over fora long time. Mathematical implication is outside time." (Peter J Eccles,"An Introduction to Mathematical Reasoning", 1997)

"Suppose the reasoning centers of the brain can get their hands on the mechanisms that plop shapes into the array and that read their locations out of it. Those reasoning demons can exploit the geometry of the array as a surrogate for keeping certain logical constraints in mind. Wealth, like location on a line, is transitive: if A is richer than B, and B is richer than C, then A is richer than C. By using location in an image to symbolize wealth, the thinker takes advantage of the transitivity of location built into the array, and does not have to enter it into a chain of deductive steps. The problem becomes a matter of plop down and look up. It is a fine example of how the form of a mental representation determines what is easy or hard to think." (Steven Pinker, "How the Mind Works", 1997)

"Statistical thinking is a general, fundamental, and independent mode of reasoning about data, variation, and chance." (David S Moore, 1998)

"For most problems found in mathematics textbooks, mathematical reasoning is quite useful. But how often do people find textbook problems in real life? At work or in daily life, factors other than strict reasoning are often more important. Sometimes intuition and instinct provide better guides; sometimes computer simulations are more convenient or more reliable; sometimes rules of thumb or back-of-the-envelope estimates are all that is needed." (Lynn A Steen,"Twenty Questions about Mathematical Reasoning", 1999)

"Under the label 'cognitive maps', mental models have been conceived of as the mental representation of spatial aspects of the environment. A mental model, in this sense, comprises the topology of an area, including relevant districts, landmarks, and paths. [...] Under the label 'naive physics', mental models have been conceived of as the mental representation of natural or technical systems. A mental model, in this sense, comprises the effective determinants, true or not, of the functioning of a physical system. [...] Under the label 'model based reasoning', the mental models notion is featured in yet another area of cognitive science - deductive reasoning. In contrast to the commonly held view that logical competence depends on formal rules of deduction, it has been argued that reasoning is a semantic process based on the manipulation of mental models. [...] Finally, under terms like 'discourse model', 'situation model', or 'scenario', mental models have been conceived of as the mental representation of a verbal description of some real or fictional state of affairs. The role of mental models in the comprehension of discourse is discussed in more detail below." (Gert Rickheit & Lorenz Sichelschmidt, "Mental Models: Some Answers, Some Questions, Some Suggestions", 1999)

"What it means for a mental model to be a structural analog is that it embodies a representation of the spatial and temporal relations among, and the causal structures connecting the events and entities depicted and whatever other information that is relevant to the problem-solving talks. […] The essential points are that a mental model can be nonlinguistic in form and the mental mechanisms are such that they can satisfy the model-building and simulative constraints necessary for the activity of mental modeling." (Nancy J Nersessian, "Model-based reasoning in conceptual change", 1999)

🪷On Mind: On Reasoning (1980-1989)

"The advantage of semantic networks over standard logic is that some selected set of the possible inferences can be made in a specialized and efficient way. If these correspond to the inferences that people make naturally, then the system will be able to do a more natural sort of reasoning than can be easily achieved using formal logical deduction." (Avron Barr, Natural Language Understanding, AI Magazine Vol. 1 (1), 1980)

"Perhaps the best way to approach the question of what mathematics is, is to start at the beginning. In the far distant prehistoric past, where we must look for the beginnings of mathematics, there were already four major faces of mathematics. First, there was the ability to carry on the long chains of close reasoning that to this day characterize much of mathematics. Second, there was geometry, leading through the concept of continuity to topology and beyond. Third, there was number, leading to arithmetic, algebra, and beyond. Finally there was artistic taste, which plays so large a role in modern mathematics. There are, of course, many different kinds of beauty in mathematics. In number theory it seems to be mainly the beauty of the almost infinite detail; in abstract algebra the beauty is mainly in the generality. Various areas of mathematics thus have various standards of aesthetics." (Richard Hamming, "The Unreasonable Effectiveness of Mathematics", The American Mathematical Monthly Vol. 87" (2), 1980)

"The advantage of semantic networks over standard logic is that some selected set of the possible inferences can be made in a specialized and efficient way. If these correspond to the inferences that people make naturally, then the system will be able to do a more natural sort of reasoning than can be easily achieved using formal logical deduction." (Avron Barr, Natural Language Understanding, AI Magazine Vol. 1" (1), 1980)

"The invalid assumption that correlation implies cause is probably among the two or three most serious and common errors of human reasoning." (Stephen J Gould, "The Mismeasure of Man", 1980)

"A person who thinks by images becomes less and less capable of thinking by reasoning, and vice versa. The intellectual process based on images is contradictory to the intellectual process of reasoning that is related to the word. There are two different ways of dealing with an object. They involve not only different approaches, but even more important, opposing mental attitudes. This is not a matter of complementary processes, such as analysis and synthesis or logic and dialectic. These processes lack any qualitative common denominator." (Jacques Ellul, "The Humiliation of the Word", 1981) 

"All advances of scientific understanding, at every level, begin with a speculative adventure, an imaginative preconception of what might be true - a preconception that always, and necessarily, goes a little way" (sometimes a long way) beyond anything which we have logical or factual authority to believe in. It is the invention of a possible world, or of a tiny fraction of that world. The conjecture is then exposed to criticism to find out whether or not that imagined world is anything like the real one. Scientific reasoning is therefore at all levels an interaction between two episodes of thought - a dialogue between two voices, the one imaginative and the other critical; a dialogue, as I have put it, between the possible and the actual, between proposal and disposal, conjecture and criticism, between what might be true and what is in fact the case." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"Scientific theories" (I have said) begin as imaginative constructions. The begin, if you like, as stories, and the purpose of the critical or rectifying episode in scientific reasoning is precisely to find out whether or not these stories are stories about real life. Literal or empiric truthfulness is not therefore the starting-point of scientific enquiry, but rather the direction in which scientific reasoning moves. If this is a fair statement, it follows that scientific and poetic or imaginative accounts of the world are not distinguishable in their origins. They start in parallel, but diverge from one another at some later stge. We all tell stories, but the stories differ in the purposes we expect them to fulfil and in the kinds of evaluations to which they are exposed." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"A mental model is a collection of 'connected' autonomous objects. Running  a mental model corresponds to modifying the parameters of the model by propagating information using the internal rules and specified topology. Running a mental model can also occur when autonomous objects change state. For us the definition of state is distinct from the current parameter values of an object. A state change consists of the replacement of one set of behavior rules with another." (Michael D Williams et al, "Human Reasoning About a Simple Physical System", [in "Mental Models", Ed(s). Dedre Gentner & Albert L Stevens], 1983)

"Since mental models can take many forms and serve many purposes, their contents are very varied. They can contain nothing but tokens that represent individuals and identities between them, as in the sorts of models that are required for syllogistic reasoning. They can represent spatial relations between entities, and the temporal or causal relations between events. A rich imaginary model of the world can be used to compute the projective relations required for an image. Models have a content and form that fits them to their purpose, whether it be to explain, to predict, or to control." (Philip Johnson-Laird, "Mental models: Toward a cognitive science of language, inference, and consciousness", 1983)

“[…] mathematics is not just a symbolism, a set of conventions for the use of special, formal vocabularies, but is intimately connected with the structure of rational thought, with reasoning practices. [...] mathematics is not just a language, and of refusing the foundationalist move of trying to reduce mathematics to logic, instead seeing mathematics as providing rational frameworks for science, is to set science against a background of rational structures and rational methods which itself has a built-in dynamics. The rational framework of science is itself historically conditioned, for it changes with developments in mathematics." (Mary Tiles,"Bachelard: Science and Objectivity", 1984)

"Concepts are inventions of the human mind used to construct a model of the world. They package reality into discrete units for further processing, they support powerful mechanisms for doing logic, and they are indispensable for precise, extended chains of reasoning. […] A mental model is a cognitive construct that describes a person's understanding of a particular content domain in the world." (John Sown,"Conceptual Structures: Information Processing in Mind and Machine", 1984)

"Mathematical rigor is the clarification of the reasoning used in mathematics. Usually, mathematics first arises in some particular situation, and as the demand for rigor becomes apparent more careful definitions of what is being reasoned about are required, and a closer examination of the numerous 'hidden assumptions' is made." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"For generations, scientists and philosophers have tried to explain ordinary reasoning in terms of logical principles - with virtually no success. I suspect this enterprise failed because it was looking in the wrong direction: common sense works so well not because it is an approximation of logic; logic is only a small part of our great accumulation of different, useful ways to chain things together." (Marvin Minsky, "The Society of Mind", 1987) 

"One of the features that distinguishes applied mathematics is its interest in framing important questions about the observed world in a mathematical way. This process of translation into a mathematical form can give a better handle for certain problems than would be otherwise possible. We call this the modeling process. It combines formal reasoning with intuitive insights. Understanding the models devised by others is a first step in learning some of the skills involved, and that is how we proceed in this text, which is an informal introduction to the mathematics of dynamical systems." (Edward Beltrami,"Mathematics for Dynamic Modeling", 1987)"

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

"One of the features that distinguishes applied mathematics is its interest in framing important questions about the observed world in a mathematical way. This process of translation into a mathematical form can give a better handle for certain problems than would be otherwise possible. We call this the modeling process. It combines formal reasoning with intuitive insights. Understanding the models devised by others is a first step in learning some of the skills involved, and that is how we proceed in this text, which is an informal introduction to the mathematics of dynamical systems." (Edward Beltrami, "Mathematics for Dynamic Modeling", 1987)

"Despite the prevailing use of graphs as metaphors for communicating and reasoning about dependencies, the task of capturing informational dependencies by graphs is not at all trivial." (Judea Pearl, "Probabilistic Reasoning in Intelligent Systems: Network of Plausible, Inference", 1988)

“Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems - general and specific statements - can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences." (Sir Erik C Zeeman,"Private Games", 1988) 

"Formal logic and the logical syllogism encapsulate connectedness in reasoning." (Marshall McLuhan & Eric McLuhan, "Laws of Media: The New Science", 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", [Noble lecture] 1988)

"Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems - general and specific statements - can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences." (Sir Erik C Zeeman, "Private Games", 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", [Noble lecture] 1988)

⏳On History of Mathematics (2010-)

"The history of mathematical ideas is often very difficult to untangle, and because ideas evolve gradually over a long period of time it is impossible to draw an exact boundary between a given theory and its offspring. It is consequently a painful task to credit some developments to a small number of authors." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"The most amazing event in the history of Greek mathematics has to have been the discovery of irrational numbers. This was not merely a fact about real numbers, which didn’t exist yet. It was a blow to Pythagorean philosophy, one of the main tenets of which was that all was number and all relations were thus ratios. And it was a genuine foundational crisis: the discovery of irrational numbers invalidated mathematical proofs. More than that, it left open the question of what one even meant by proportion and similarity." (Craig Smoryński, "History of Mathematics: A Supplement", 2008)

"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 history of mathematics can be studied chronologically, thematically, topically, and biographically. I have used in this course elements of each approach." (Israel Kleiner, Excursions in the History of Mathematics", 2012)

"The issue of rigorous foundations for calculus began with gropings in the early seventeenth century and concluded with a 'final' resolution in the 1870s. This rather slow evolution toward a logical grounding is not atypical in the history of mathematics. Rigor, formalism, and the logical development of a concept, result, or theory usually come at the end of a process of mathematical evolution. In the case of calculus, mathematicians achieved very impressive results during the seventeenth and eighteenth centuries by intuitive, heuristic reasoning, and therefore had no compelling reasons to put their subject on firm foundations. This does not mean that there was no concern during these two centuries for the logic behind the algorithms of calculus; and there were attempts, albeit unsuccessful, to supply it." (Israel Kleiner, Excursions in the History of Mathematics", 2012)

"The early history of group theory and Galois theory are closely related - after all, Galois was the person who introduced the term 'group' into mathematics. So it is not surprising that notions like Abelian equations from Galois theory influenced the terminology of group theory." 

"When a mathematical conjecture eventually turns out to be correct, its history often follows a standard pattern. Over a period of time, various people prove the conjecture to be true provided special restrictions apply. Each such result improves on the previous one by relaxing some restrictions, but eventually this process runs out of steam. Finally, a new and much cleverer idea completes the proof." (Ian Stewart, "Visions of Infinity", 2013)

"The history of mathematical ideas is often very difficult to untangle, and because ideas evolve gradually over a long period of time it is impossible to draw an exact boundary between a given theory and its offspring. It is consequently a painful task to credit some developments to a small number of authors." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"History of mathematics is done by mathematicians as well as historians. History models mathematics as a segment of the ongoing story of human culture. Mathematicians are likely to see the past through the eyes of the present, and ask, ‘Was it important? natural? deep? surprising? elegant?’ The historian sees mathematics as a thread in the ever-growing web of human life, intimately interwoven with finance and technology, with war and peace. Today's mathematics is the culmination of all that has happened before now, yet to future viewpoints it will seem like a brief, outmoded stage of the past." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)

⏳About Mathematicians (-1699)

"I think that those concerned with the sciences [mathemata] are men of discernment, and it is not strange that they should think correctly about the nature of particular things. And so they have handed down to us clear knowledge of the speed of the heavenly bodies and their risings and settings, of geometry, numbers and, not least, of the science of music. For these sciences seem to be related: they are concerned with the first two kinds of what is, which are related." (Archytas, cca 5th century)

"Therefore I would not have it unknown to Your Holiness, the the only thing which induced me to look for another way of reckoning the movements of the heavenly bodies was that I knew that mathematicians by no means agree in their investigation thereof." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543

"What has philosophy got to do with measuring anything? It's the mathematicians you have to trust, and they measure the skies like we measure a field." (Galileo Galilei, "Concerning the New Star", 1606)

"A mathematician, as good as he may be, without the support of a good drawing, is nothing but a half-mathematician, but also a man without eyes." (Lodovico Cardi, [letter to Galileo Galilei] 1611)

"Grant a mathematician but one minute principle, he immediately draws a consequence from it, to which you must necessarily assent; and from this consequence another, till he leads you so far (whether you will or no) that you have much ado to believe all he has proved, and what you have already assented to." (Bernard Le Bovier de Fontenelle, "Conversations on the Plurality of Worlds", 1686)

🅲Nicolaus Copernicus - Collected Quotes

"The center of the earth is not the center of the universe, but only of gravity and of the lunar sphere." (Nicolaus Copernicus, "Commentariolus" [Little Commentary"], cca. 1514)

"Although all the good arts serve to draw man's mind away from vices and lead it toward better things, this function can be more fully performed by this art, which also provides extraordinary intellectual pleasure." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"Among the many and varied literary and artistic studies upon which the natural talents of man are nourished, I think that those above all should be embraced and pursued with the most loving care which have to do with things that are very beautiful and very worthy of knowledge." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"At rest, however, in the middle of everything is the sun. For, in this most beautiful temple, who would place this lamp in another or better position than that from which it can light up the whole thing at the same time?" (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"But we should rather follow the wisdom of nature, which, as it takes very great care not to have produced anything superfluous or useless, often prefers to endow one thing with many effects.Books & Literature." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"For a traveler going from any place toward the north, that pole of the daily rotation gradually climbs higher, while the opposite pole drops down an equal amount." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"For it is the duty of an astronomer to compose the history of the celestial motions through careful and expert study." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"For when a ship is floating calmly along, the sailors see its motion mirrored in everything outside, while on the other hand they suppose that they are stationary, together with everything on board. In the same way, the motion of the earth can unquestionably produce the impression that the entire universe is rotating." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"Hence I feel no shame in asserting that this whole region engirdled by the moon, and the center of the earth, traverse this grand circle amid the rest of the planets in an annual revolution around the sun. Near the sun is the center of the universe. Moreover, since the sun remains stationary, whatever appears as a motion of the sun is really due rather to the motion of the earth." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"I shall now recall to mind that the motion of the heavenly bodies is circular, since the motion appropriate to a sphere is rotation in a circle." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"[...] if the worth of the arts were measured by the matter with which they deal, this art - which some call astronomy, others astrology, and many of the ancients the consummation of mathematics - would be by far the most outstanding. This art which is as it were the head of all the liberal arts and the one most worthy of a free man leans upon nearly all the other branches of mathematics. Arithmetic, geometry, optics, geodesy, mechanics, and whatever others, all offer themselves in its service." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"More stars in the north are seen not to set, while in the south certain stars are no longer seen to rise." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"Of all things visible, the highest is the heaven of the fixed stars." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"The earth together with its surrounding waters must in fact have such a shape as its shadow reveals, for it eclipses the moon with the arc of a perfect circle." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"The earth also is spherical, since it presses upon its center from every direction." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"The massive bulk of the earth does indeed shrink to insignificance in comparison with the size of the heavens." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"The strongest affection and utmost zeal should, I think, promote the studies concerned with the most beautiful objects. This is the discipline that deals with the universe's divine revolutions, the stars' motions, sizes, distances, risings and settings... for what is more beautiful than heaven?" (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"Therefore I would not have it unknown to Your Holiness, the the only thing which induced me to look for another way of reckoning the movements of the heavenly bodies was that I knew that mathematicians by no means agree in their investigation thereof." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543) 

"Those who devised the eccentrics seen thereby in large measure to have solved the problem of apparent motions with approximate calculations. But meanwhile they introduced a good many ideas which apparently contradict the first principles of uniform motion. Nor could they elicit or deduce from the eccentrics the principal consideration, that is, the structure of the universe and the true symmetry of its parts." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"We find then in this arrangement an admirable harmony of the world, and a dependable, harmonious interconnexion of the motion and the size of the paths, such as otherwise cannot be discovered. For here the penetrating observer can note why the forward and the retrograde movement of Jupiter appears greater than that of Saturn, and smaller than that of Mars, and again greater with Venus than with Mercury; and why such retrogression appears oftener with Saturn than with Jupiter, less often with Mars and Venus than with Mercury. Moreover, why Saturn, Jupiter, and Mars, when they rise in the evening, appear greater than when they disappear and reappear [with the sun] [...]And all this results from the same cause, namely the motion of the earth." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"We regard it as a certainty that the earth, enclosed between poles, is bounded by a spherical surface." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543) 

"Whatever motion appears in the firmament arises not from any motion of the firmament, but from the earth's motion. The earth together with its circumjacent elements performs a complete rotation on its fixed poles in a daily motion, while the firmament and highest heaven abide unchanged." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres" ["De revolutionibus orbium coelestium"], 1543)

"And as far as hypotheses go, let no one expect anything in the way of certainty from astronomy, since astronomy can offer us nothing certain, lest, if anyone take as true that which has been constructed for another use, he go away from this discipline a bigger fool than when he came to it." (Andreas Osiander, "Ad lectorem de hypothesibus huius operis" [To the Reader Concerning the Hypotheses of this Wor"], [in Nicolaus Copernicus' "On the Revolutions of the Heavenly Spheres", 1543], 1943)

📒🆆John Wyndham - Collected Quotes

"The humans have a curious force they call ambition. It drives them, and, through them, it drives us. This force which keeps them active, we lack. Perhaps, in time, we machines will acquire it." (John Wyndham, "The Lost Machine", 1932)

"There are so many disadvantages in human construction which do not occur in us machines. [...] Some little thing here or there breaks - they stop working and then, in a short time, they are decomposing. Had he been a machine, like myself, I could have mended him, replaced the broken parts and made him as good as new, but with these animal structures one is almost helpless." (John Wyndham, "The Lost Machine", 1932)

"And we danced, on the brink of an unknown future, to an echo from a vanished past." (John Wyndham, "The Lost Machine", 1932)

"I don't think it had ever occurred to me that man's supremacy is not primarily due to his brain, as most of the books would have one think. It is due to the brain's capacity to make use of the information conveyed to it by a narrow band of visible light rays. His civilization, all that he had achieved or might achieve, hung upon his ability to perceive that range of vibrations from red to violet. Without that, he was lost." (John Wyndham, "The Day of the Triffids", 1951)

"There is an inability to sustain the tragic mood, a phoenix quality of the mind. It may be helpful or harmful, it is just a part of the will to survive - yet, also, it has made it possible for us to engage in one weakening war after another." (John Wyndham, "The Day of the Triffids", 1951)

"Knowing makes all the difference... It's the difference between just trying to keep alive, and having something to live for." (John Wyndham, "The Chrysalids", 1955)

"The essential quality of life is living, the essential quality of living is change; change is evolution; and we are part of it." (John Wyndham, "The Chrysalids", 1955)

"But, as I understand it, your God is a universal God; He is God on all suns and all planets. Surely, then, He must have universal form? Would it not be a staggering vanity to imagine that He can manifest Himself only in the form that is appropriate to this particular, not very important planet?" (John Wyndham, "The Midwich Cuckoos" 1957)

"Knowledge is simply a kind of fuel; it needs the motor of understanding to convert it into power." (John Wyndham, "The Midwich Cuckoos" 1957)

"Some quotations are greatly improved by lack of context." (John Wyndham, "The Midwich Cuckoos" 1957) 

🅿Louis Pasteur - Collected Quotes

"In the fields of observation chance favors only the prepared mind." (Louis Pasteur, [lecture] 1854)

"As in the experimental sciences, truth cannot be distinguished from error as long as firm principles have not been established through the rigorous observation of facts." (Louis Pasteur, "Étude sur la maladie des vers à soie", 1870)

"Analogy cannot serve as proof." (Louis Pasteur)

"Man’s first glance at the universe discovers only variety, diversity, multiplicity of phenomena. Let that glance be illuminated by science - by the science which brings man closer to God, - and simplicity and unity shine on all sides." (Louis Pasteur)

"The arguments […] by which you support my theories, are most ingenious, but not founded on demonstrated facts; analogy is no proof." (Louis Pasteur)

 "[…] the notion of the infinite […] forces itself upon our mind and yet is incomprehensible. When this notion takes possession of the understanding we have only to bow before it." (Louis Pasteur)

🅿Pythagoras of Samos - Collected Quotes

"All was numbers." (Pythagoras of Samos, cca. 6th century BC)

"Geometry is knowledge of the eternally existent."  (Pythagoras of Samos, cca. 6th century BC)

"Number is the ruler of forms and ideas, and the cause of gods and demons." (Pythagoras of Samos, cca. 6th century BC)

"Number rules the universe." (Pythagoras of Samos, cca. 6th century BC)

"Number was the substance of all things." (Pythagoras of Samos, cca. 6th century BC)

"The learning of many things does not teach intelligence […]." (Pythagoras of Samos, cca. 6th century BC)

"There is geometry in the humming of the strings; there is music in the spacing of the spheres." (Pythagoras of Samos, cca. 6th century BC)

🅿Edgar E Peters - Collected Quotes

"Because we hate living with uncertainty, we often try to make complex systems, such as the economy, more predictable and less uncertain. However, making a complex system more predictable also makes it less resilient to shocks, and less creative. Lowering uncertainty reduces complexity, often with disastrous effects." (Edgar E Peters, "Patterns in the dark: understanding risk and financial crisis with complexity theory", 1999)

"Complex systems, then, have local uncertainty and global certainty. They generate change, and they are resilient to unexpected shocks. They turn uncertainty into order, and they reverse order back into uncertainty. They evolve and change through time, and they do so without a central planner. Complex systems are everywhere. In fact, real life is one huge complex system. How is such behavior possible? First, we need to understand the general class of complex systems. By understanding their nature, we will see the important role uncertainty plays in maintaining stability. When we understand natural systems, we will understand the role of uncertainty in a free society." (Edgar E Peters, "Patterns in the dark: understanding risk and financial crisis with complexity theory", 1999)

In a free-market economy, then, uncertainty is a necessary element. Only when the economy is in a state of uncertainty can the participants efficiently search for solutions to problems and find creative answers. In addition, only a system that depends on uncertainty can survive unexpected shocks. A complex process can take multiple paths to an optimal solution. It does not require 'ideal' conditions; in fact, shocks often force it to find a better solution, a higher hill in the fitness landscape. The 'creative destruction' identified by the Austrian school suggests that a free-market economy is not only resilient to shocks, but is also creative and capable of generating innovation. It can only do so while in a high state of uncertainty." (Edgar E Peters, "Patterns in the dark: understanding risk and financial crisis with complexity theory", 1999)

"It seems obvious that uncertainty reigns during times of crisis. However, crisis itself can be a positive development. It is only negative in that it specifies that change is coming. Most people are uncomfortable with change and equate it with hard times. Mainstream economics tends to take the same view, calling such events 'shocks'. They are even referred to as 'exogenous' - outside of the system. If it were not for change, according to the mainstream school, everything would continue along in perfect balance, a 'circular flow'. Change is like an alien invasion. The mainstream view ignores the fact that change is necessary." (Edgar E Peters, "Patterns in the dark: understanding risk and financial crisis with complexity theory", 1999)

"Uncertainty is not necessarily bad or synonymous with risk. Complex systems use uncertainty to their advantage as they adapt to changes in their environment and learn to be resilient to unexpected shocks. Uncertainty then, rather than being the source of so many problems, becomes a necessary element if a market and a society are to remain free." (Edgar E Peters, "Patterns in the dark: understanding risk and financial crisis with complexity theory", 1999) 

🅿John Napier - Collected Quotes

"Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers. [...] I began therefore to consider in my mind by what certain  and ready art I might remove those hindrances." (John Napier, "Mirifici logarithmorum canonis descriptio", 1614)

"A Logarithmic Table is a small table by the use of which we can obtain a knowledge of all geometrical dimensions and motions in space, by a very easy calculation. It is deservedly called very small, because it does not exceed in size a table of sines; very easy, because by it all multiplications, divisions, and the more difficult extractions of roots are avoided; for by only a very few most easy additions, subtractions, and divisions by two, it measures quite generally all figures and motions."  (John Napier, "The Construction of the Wonderful Canon of Logarithms", 1889)

"And if any number of equals to a first sine be multiplied together producing a second, just so many equals to the Logarithm of the first added together produce the Logarithm of the second." (John Napier, "The Construction of the Wonderful Canon of Logarithms", 1889)

"Any desired geometrical mean between two sines has for its Logarithm the corresponding arithmetical mean between the Logarithms of the sines." (John Napier, "The Construction of the Wonderful Canon of Logarithms", 1889)

"To decrease geometrically is this, that in equal times, first the whole quantity then each of its successive remainders is diminished, always by a like proportional part." (John Napier, "The Construction of the Wonderful Canon of Logarithms", 1889)