⏳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)

"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)

"Knowledge is simply a kind of fuel; it needs the motor of understanding to convert it into power." (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)

🪷On Mind: On Reasoning (1970-1979)

"At root what is needed for scientific inquiry is just receptivity to data, skill in reasoning, and yearning for truth. Admittedly, ingenuity can help too." (Willard v O Quine, "The Web of Belief", 1970)

"All advances of scientific understanding, at every level, begin with a speculative adventure, an imaginative preconception of what might be true.[...] [This] 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 [...]" (Sir Peter B Medawar,  "The Hope of Progress", 1972)

"Confidence in the omnicompetence of statistical reasoning grows by what it feeds on." (Harry Hopkins, "The Numbers Game: The Bland Totalitarianism", 1973)

"Heuristic reasoning is good in itself. What is bad is to mix up heuristic reasoning with rigorous proof. What is worse is to sell heuristic reasoning for rigorous proof." (George Pólya, "How to Solve It", 1973)

"When we can’t prove our point through the use of sound reasoning, we fall back upon statistical ‘mumbo jumbo’ to confuse and demoralize our opponents." (Audrey Haber & Richard P Runyon, "General Statistics", 1973)

"All perceiving is also thinking, all reasoning is also intuition, all observation is also invention." (Rudolf Arnheim, "Entropy and Art: An Essay on Disorder and Order", 1974) 

"Demonstrative reasoning differs from plausible reasoning just as the fact differs from the supposition, just as actual existence differs from the possibility of existence. Demonstrative reasoning is reliable, incontrovertible and final. Plausible reasoning is conditional, arguable and oft-times risky." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"Every science is permeated with proofs, and to the same extent as mathematics, for demonstrative reasoning is an integral part of mathematics." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"In mathematics the problem of the essence of proof has been thoroughly worked out and every mathematician must master the methods of demonstrative reasoning. Appropriate rules have been established for this purpose. These rules and the concepts of rigour and exactitude of reasoning vary from century to century, and at the present time every mathematician knows the level of rigour of modern mathematics." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"In plausible reasoning, one must distinguish a reasonable conjecture from a less reasonable conjecture and be able to substantiate the conjecture with the available facts, to find these facts, to search painstakingly for facts that contradict the conjecture, and to correlate the findings and again return to plausible arguments." (Yakov Khurgin, "Did You Say Mathematics?", 1974) 

"In reasoning, as in every other activity, it is, of course, easy to fall into error. In order to reduce this risk, at least to some extent, it is useful to support intuition with suitable superstructures: in this case, the superstructure is logic" (or, to be precise, the logic of certainty)." (Bruno de Finetti, "Theory of Probability", 1974)

"Mathematical knowledge is fixed securely by means of demonstrative reasoning, but the approaches to such knowledge are strewn with plausible modes of reasoning." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"Mathematics is the sole avenue for learning how to reason via proof. On the other hand, one must also learn how to conjecture.[…] In a rigorous case of demonstrative reasoning, the main thing is to be able to distinguish proof from conjecture, justified proof from an unjustified attempt." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"Most people imagine mathematics to be a deductive science in which all theorems, results and facts are obtained via logical reasoning by proceeding from certain starting axioms, primal assertions, assumed to be self-evident or not requiring any proof." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"[Fuzzy logic is] a logic whose distinguishing features are (1) fuzzy truth-values expressed in linguistic terms, e. g., true, very true, more or less true, or somewhat true, false, nor very true and not very false, etc.; (2) imprecise truth tables; and (3) rules of inference whose validity is relative to a context rather than exact." (Lotfi A Zadeh, "Fuzzy logic and approximate reasoning", 1975)

"I find it more difficult, but also much more fun, to get the right answer by indirect reasoning and before all the evidence is in. It’s what a theoretician does in science. But the conclusions drawn in this way are obviously more risky than those drawn by direct measurement, and most scientists withhold judgment until there is more direct evidence available. The principal function of such detective work - apart from entertaining the theoretician - is probably to so annoy and enrage the observationalists that they are forced, in a fury of disbelief, to perform the critical measurements." (Carl Sagan, "The Cosmic Connection: An Extraterrestrial Perspective", 1975)

"What science and the quest for knowledge are after is irrefutable truth; that is, propositions that human beings are not free to reject — that are compelling. They are of two kinds, as we have known since Leibnitz: truths of reasoning and truths of fact." (Hannah Arendt, The New Yorker, 1977)

"[…] the use of analogies, particularly with metaphor, adds richness and dimension to arguments and descriptions not possible with ordinary discourse or with propositional reasoning." (Jeanette M Gallagher,"The Future of Formal Thought Research: The Study of Analogy and Metaphor", 1978)

"All mathematical problems are solved by reasoning within a deductive system in which basic laws of logic are embedded." (Martin Gardner, "Aha! Insight", 1978)

"Many people believe that reasoning, and therefore science, is a different activity from imagining. But this is a fallacy […] Reasoning is constructed with movable images just as certainly as poetry is." (Jacob Bronowski, "Visionary Eye", 1978)

"Mathematical Reasoning is not only exact; it has its own criteria of reality." (Paul K Feyerabend,"Science in a Free Society", 1978)

"Reasoning is constructed with movable images just as certainly as poetry is." (Jacob Bronowski, "Visionary Eye", 1978)

🪷On Mind: On Reasoning (1960-1969)

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

"Today we preach that science is not science unless it is quantitative. We substitute correlation for causal studies, and physical equations for organic reasoning. Measurements and equations are supposed to sharpen thinking, but [...] they more often tend to make the thinking non-causal and fuzzy." (John R Platt, "Strong Inference", Science Vol. 146" (3641), 1964) 

“Mathematics is a form of poetry which transcends poetry in that it proclaims a truth; a form of reasoning which transcends reasoning in that it wants to bring about the truth it proclaims; a form of action, of ritual behavior, which does not find fulfilment in the act but must proclaim and elaborate a poetic form of truth." (Salomon Bochner,"Why Mathematics Grows", Journal of the History of Ideas, 1965)

"So the first thing we have to accept is that even in mathematics you can start in different places. If all these various theorems are interconnected by reasoning there is no real way to say ‘These are the most fundamental axioms’, because if you were told something different instead you could also run the reasoning the other way. It is like a bridge with lots of members, and it is over-connected; if pieces have dropped out you can reconnect it another way." (Richard Feynman, "The Character of Physical Law", 1965)

"When the problems in physics become difficult we may often look to the mathematician who may already have studied such things and have prepared a line of reasoning for us to follow. On the other hand they may not have, in which case we have to invent our own line of reasoning, which we then pass back to the mathematician." (Richard Feynman,"The Character of Physical Law" , 1965) 

"Analogy serves to provoke certain types of questions which can, on investigation, lead to the recognition of more comprehensive ranges of order in the archaeological data." (Lewis R Binford, "Smudge Pits and Hide Smoking: The Use of Analogy in Archaeological Reasoning, American Antiquity Vol. 32" (1), 1967)

"But, really, mathematics is not religion; it cannot be founded on faith. And what was most important, the methods yielding such remarkable results in the hands of the great masters began to lead to errors and paradoxes when employed by their less talented students. The masters were kept from error by their perfect mathematical intuition, that subconscious feeling that often leads to the right answer more quickly than lengthy logical reasoning. But the students did not possess this intuition […]"(Naum Ya. Vilenkin, "Stories about Sets", 1968) "

"Infinite sets possess remarkable properties. In studying these properties mathematicians were led to continually perfect their reasoning and to further develop mathematical logic." (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"There is no substitute for honest, thorough, scientific effort to get correct data" (no matter how much it clashes with preconceived ideas). There is no substitute for actually reaching a correct chain of reasoning. Poor data and good reasoning give poor results. Good data and poor reasoning give poor results. Poor data and poor reasoning give rotten results." (Edmund C Berkeley, Computers and Automation, 1969)

🪷On Mind: On Reasoning (1950-1959)

"It has been said, often enough and certainly with good reason, that teaching mathematics affords a unique opportunity to teach demonstrative reasoning. I wish to add that teaching mathematics also affords an excellent opportunity to teach plausible reasoning. A student of mathematics should learn, of course, demonstrative reasoning; it is his profession and the distinctive mark of his science. Yet he should also learn plausible reasoning; this is the kind of reasoning on which his creative work will mainly depend, The general student should get a taste of demonstrative reasoning; he may have little opportunity to use it directly, but he should acquire a standard with which he can compare alleged evidence of all sorts aimed at him in modern life. He needs, however, in all his endeavors plausible reasoning. At any rate, an ambitious teacher of mathematics should teach both kinds of reasoning to both kinds of students." (George Pólya, "On Plausible Reasoning", Proceedings of the International Congress of Mathematics, 1950)

"Demonstrative reasoning is safe, beyond controversy, and final. Plausible reasoning is hazardous, controversial, and provisional. Demonstrative reasoning penetrates the sciences just as far as mathematics does, but it is in itself" (as mathematics is in itself) incapable of yielding essentially new knowledge about the world around us. Anything new that we learn about the world involves plausible reasoning, which is the only kind of reasoning, for which we care in everyday affairs. Demonstrative reasoning has rigid standards, codified and clarified by logic" (formal or demonstrative logic), which is the theory of demonstrative reasoning. The standards of plausible reasoning are fluid, and there is no theory of such reasoning that could be compared to demonstrative logic in clarity or would command comparable consensus." (George Pólya, "Mathematics and Plausible Reasoning", 1954)

"Demonstrative reasoning penetrates the sciences just as far as mathematics does, but it is in itself" (as mathematics is in itself) incapable of yielding essentially new knowledge about the world around us. Anything new that we learn about the world involves plausible reasoning, which is the only kind of reasoning for which we care in everyday affairs." (George Pólya, "Induction and Analogy in Mathematics", 1954)

"From the outset it was clear that the two kinds of reasoning have different tasks. From the outset. they appeared very different: demonstrative reasoning as definite, final, 'machinelike'; and plausible reasoning as vague, provisional, specifically 'human'. Now we may see the difference a little more distinctly. In opposition to demonstrative inference, plausible inference leaves indeterminate a highly relevant point: the 'strength' or the 'weight' of the conclusion. This weight may depend not only on clarified grounds such as those expressed in the premises, hut also on unclarified unexpressed grounds somewhere on the background of the person who draws the conclusion. A person has a background, a machine has not. Indeed, you can build a machine to draw demonstrative conclusions for you, but I think you can never build a machine that will draw plausible inferences." (George Pólya, "Mathematics and Plausible Reasoning", 1954)

"If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it." (George Pólya, "Mathematics and plausible reasoning" Vol. 1, 1954)

"In plausible reasoning the principal thing is to distinguish... a more reasonable guess from a less reasonable guess." (George Pólya, "Mathematics and plausible reasoning" Vol. 1, 1954)

"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference." (George Pólya, "Mathematics and plausible reasoning" Vol. 1, 1954)

"We secure our mathematical knowledge by demonstrative reasoning, but we support our conjectures by plausible reasoning. A mathematical proof is demonstrative reasoning, but the inductive evidence of the physicist, the circumstantial evidence of the lawyer, the documentary evidence of the historian, and the statistical evidence of the economist belong to plausible reasoning." (George Pólya, "Mathematics and Plausible Reasoning", 1954)

"You have to guess the mathematical theorem before you prove it: you have to guess the idea of the proof before you carry through the details. You have to combine observations and follow analogies: you have to try and try again. The result of the mathematician’s creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing" (George Pólya, "Mathematics and plausible reasoning" Vol. 1, 1954)

"Science cannot be based on dogma or authority of any kind, nor on any institution or revelation, unless indeed it be of the Book of Nature that lies open before our eyes. We need not dwell on the processes of acquiring knowledge by observation, experiment, and inductive and deductive reasoning. The study of scientific method both in theory and practice is of great importance. It is inherent in the philosophy that the record may be imperfect and the conceptions erroneous; the potential fallibility of our science is not only acknowledged but also insisted upon." (Sir Robert Robinson, "Science and the Scientist", Nature Vol. 176" (4479), 1955)

"The following are some aspects of the artificial intelligence problem: […] If a machine can do a job, then an automatic calculator can be programmed to simulate the machine. […] It may be speculated that a large part of human thought consists of manipulating words according to rules of reasoning and rules of conjecture. From this point of view, forming a generalization consists of admitting a new word and some rules whereby sentences containing it imply and are implied by others. This idea has never been very precisely formulated nor have examples been worked out. […] How can a set of" (hypothetical) neurons be arranged so as to form concepts. […] to get a measure of the efficiency of a calculation it is necessary to have on hand a method of measuring the complexity of calculating devices which in turn can be done. […] Probably a truly intelligent machine will carry out activities which may best be described as self-improvement. […] A number of types of 'abstraction' can be distinctly defined and several others less distinctly. […] the difference between creative thinking and unimaginative competent thinking lies in the injection of a some randomness. The randomness must be guided by intuition to be efficient." (John McCarthy et al, "A Proposal for the Dartmouth Summer Research Project on Artificial Intelligence", 1955)

“Because intuition turned out to be deceptive in so many instances, and because propositions that had been accounted true by intuition were repeatedly proved false by logic, mathematicians became more and more skeptical of intuition. [….] Thus, a demand arose for the expulsion of intuitive reasoning and for the complete formalization of mathematics." (Hans Hahn,"The crisis in intuition", 1956) 

"Nevertheless, there are three distinct types of paradoxes which do arise in mathematics. There are contradictory and absurd propositions, which arise from fallacious reasoning. There are theorems which seem strange and incredible, but which, because they are logically unassailable, must be accepted even though they transcend intuition and imagination. The third and most important class consists of those logical paradoxes which arise in connection with the theory of aggregates, and which have resulted in a re-examination of the foundations of mathematics." (James R Newman, "The World of Mathematics" Vol. III, 1956)

"The predictions of physical theories for the most part concern situations where initial conditions can be precisely specified. If such initial conditions are not found in nature, they can be arranged. Such arrangements are considerably easier to realize with inanimate than with animate matter, because the properties of animate matter are much more sensitive to being tampered with than inanimate matter. In particular, living tissue in vitro may behave quite differently than in situ. Controlled biological experiments are, of course, possible, but they are more difficult and their scope is more limited than that of physical experiments. For this reason, biology has had to depend to a greater extent than physics on theories of larger speculative scope, in which reasoning by imaginative analogy plays a more important role." (Anatol Rapoport, "The Search for Simplicity", 1956)

"Understanding mathematical logic, or the theory of relativity, is not an indispensable attribute of the cultured mind. But if one wishes to learn anything about these subjects, one must learn something. It is necessary to master the rudiments of the language, to practice a technique, to follow step by step a characteristic sequence of reasoning and to see a problem through from beginning to end." (James R Newman, "The World of Mathematics" Vol. I, 1956)

"The function of mathematical logic is to reveal and codify the logical processes employed in mathematical reasoning and to clarify the concepts of mathematics; it is itself a branch of mathematics, employing mathematical symbolism and technique, a branch which has developed in its entirety during the past hundred years and which in its vigor and fecundity and the power and importance of its discoveries may well claim to be in the forefront of modern mathematics." (Reuben L Goodstein, "Mathematical Logic", 1957)

"A logic machine is a device, electrical or mechanical, designed specifically for solving problems in formal logic. A logic diagram is a geometrical method for doing the same thing. […] A logic diagram is a two-dimensional geometric figure with spatial relations that are isomorphic with the structure of a logical statement. These spatial relations are usually of a topological character, which is not surprising in view of the fact that logic relations are the primitive relations underlying all deductive reasoning and topological properties are, in a sense, the most fundamental properties of spatial structures. Logic diagrams stand in the same relation to logical algebras as the graphs of curves stand in relation to their algebraic formulas; they are simply other ways of symbolizing the same basic structure." (Martin Gardner, "Logic Machines and Diagrams", 1958)

"Mathematics is a model of exact reasoning, an absorbing challenge to the mind, an esthetic experience for creators and some students, a nightmarish experience to other students, and an outlet for the egotistic display of mental power." (Morris Kline,"Mathematics and the Physical World", 1959)