🪷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)

🪷On Mind: On Reasoning (1940-1949)

"It is difficult, however, to learn all these things from situations such as occur in everyday life. What we need is a series of abstract and quite impersonal situations to argue about in which one side is surely right and the other surely wrong. The best source of such situations for our purposes is geometry. Consequently we shall study geometric situations in order to get practice in straight thinking and logical argument, and in order to see how it is possible to arrange all the ideas associated with a given subject in a coherent, logical system that is free from contradictions. That is, we shall regard the proof of each proposition of geometry as an example of correct method in argumentation, and shall come to regard geometry as our ideal of an abstract logical system. Later, when we have acquired some skill in abstract reasoning, we shall try to see how much of this skill we can apply to problems from real life." (George D Birkhoff & Ralph Beately, "Basic Geometry", 1940)

"The first step in discussing a proposition with a view to proving it true or false is to discover the earlier propositions upon which it depends. A few of these earlier propositions must remain unproved and be taken for granted, for we must have some place from which to begin our chain of reasoning. We cannot prove everything. The propositions that we take for granted without proof are called assumptions. When we criticize an argument or assert that a certain proposition is true or false, we should first discover the assumptions upon which the argument or proposition depends." (George D Birkhoff & Ralph Beately, "Basic Geometry", 1940)

"The main thing geometry gives us is the ideal of a logical system and of precise thinking, and some acquaintance with the language in which logical arguments are usually expressed. The answer to a problem in actual life can often be obtained by further investigation of the actual facts, while in geometry it can always be obtained by reasoning alone." (George D Birkhoff & Ralph Beately, "Basic Geometry", 1940))

“It is his intuition, his mystical insight into the nature of things, rather than his reasoning which makes a great scientist." (Karl R Popper,"The Open Society and Its Enemies", 1945)  

"Figures alone prove or disprove nothing. Only the conclusions drawn from the collected material can do this. And these are theoretical." (Ludwig von Mises) "Geometry is the science of correct reasoning on incorrect figures." (George Pólya, "How to Solve It", 1945)

"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", 1945)

"Heuristic reasoning is reasoning not regarded as final and strict but as provisional and plausible only, whose purpose is to discover the solution of the present problem. We are often obliged to use heuristic reasoning. We shall attain complete certainty when we shall have obtained the complete solution, but before obtaining certainty we must often be satisfied with a more or less plausible guess. We may need the provisional before we attain the final. We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building." (George Pólya, "How to Solve It", 1945)

"If scientific reasoning were limited to the logical processes of arithmetic, we should not get very far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability." (Vannevar Bush, "As We May Think", Atlantic Monthly, 1945)

"It is his intuition, his mystical insight into the nature of things, rather than his reasoning which makes a great scientist." (Karl R Popper, "The Open Society and Its Enemies", 1945)

"The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. [...] Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information." (George Pólya, "How to solve it", 1945)

"We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building." (George Pólya, "How to solve it", 1945)

"As everyone knows, it is not always easy to think. By thinking, we mean computing, reasoning, and other handling of information. By information we mean collections of ideas - physically, collections of marks that have meaning. By handling information, we mean proceeding logically from some ideas to other ideas - physically, changing from some marks to other marks in ways that have meaning." (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

"The amount of human effort needed to handle information correctly depends very much on the properties of the physical equipment expressing the information, although the laws of correct reasoning are independent of the equipment." (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

🪷On Mind: On Reasoning (1930-1939)

 "The final truth about phenomena resides in the mathematical description of it; so long as there is no imperfection in this, our knowledge is complete. We go beyond the mathematical formula at our own risk; we may find a [nonmathematical] model or picture that helps us to understand it, but we have no right to expect this, and our failure to find such a model or picture need not indicate that either our reasoning or our knowledge is at fault." (James Jeans, "The Mysterious Universe", 1930)

"In the study of ideas, it is necessary to remember that insistence on hard-headed clarity issues from sentimental feeling, as it were a mist, cloaking the perplexities of fact. Insistence on clarity at all costs is based on sheer superstition as to the mode in which human intelligence functions. Our reasonings grasp at straws for premises and float on gossamers for deductions." (Alfred N Whitehead, "Adventures of Ideas", 1933) 

"Our reasonings grasp at straws for premises and float on gossamers for deductions." (Alfred N Whitehead, "Adventures of Ideas", 1933) 

"Science is not a system of certain, or -established, statements; nor is it a system which steadily advances towards a state of finality […] And our guesses are guided by the unscientific, the metaphysical" (though biologically explicable) faith in laws, in regularities which we can uncover - discover. Like Bacon, we might describe our own contemporary science - 'the method of reasoning which men now ordinarily apply to nature' - as consisting of 'anticipations, rash and premature' and as 'prejudices'." (Karl R Popper, "The Logic of Scientific Discovery", 1934)

"Science is the attempt to discover, by means of observation, and reasoning based upon it, first, particular facts about the world, and then laws connecting facts with one another and" (in fortunate cases) making it possible to predict future occurrences." (Bertrand Russell, "Religion and Science, Grounds of Conflict", 1935)

 "Scientific Ideas can often be adequately exhibited for all the purposes of reasoning, by means of Definitions and Axioms; all attempts to reason by means of Definitions from common Notions, lead to empty forms or entire confusion." (William Whewell, "History of the Inductive Sciences from the Earliest to the Present Time", 1937)

Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgments which are not the result of conscious trains of reasoning. These judgments are often but by no means invariably correct" (leaving aside the question as to what is meant by 'correct'). Often it is possible to find some other way of verifying the correctness of an intuitive judgment. One may for instance judge that all positive integers are uniquely factorable into primes; a detailed mathematical argument leads to the same result. It will also involve intuitive judgments, but they will be ones less open to criticism than the original judgment about factorization." (Alan M Turing, "Systems of Logic Based on Ordinals", Proceedings of the London Mathematical Society Vol 45" (2), 1939)

🪷On Mind: On Reasoning (1920-1929)

“Philosophy in its old form could exist only in the absence of engineering, but with engineering in existence and daily more active and far reaching, the old verbalistic philosophy and metaphysics have lost their reason to exist. They were no more able to understand the ‘production’ of the universe and life than they are now able to understand or grapple with 'production' as a means to provide a happier existence for humanity. They failed because their venerated method of ‘speculation’ can not produce, and its place must be taken by mathematical thinking. Mathematical reasoning is displacing metaphysical reasoning. Engineering is driving verbalistic philosophy out of existence and humanity gains decidedly thereby." (Alfred Korzybski, "Manhood of Humanity", 1921)

"[…] mathematics is not, never was, and never will be, anything more than a particular kind of language, a sort of shorthand of thought and reasoning. The purpose of it is to cut across the complicated meanderings of long trains of reasoning with a bold rapidity that is unknown to the mediaeval slowness of the syllogisms expressed in our words." (Charles Nordmann, "Einstein and the Universe", 1922)

"The question whether any branch of science can ever become purely deductive is easily answered. It cannot. If science deals with the external world, as we believe it does, and not merely with the relations of propositions then no branch of science can ever be purely deductive. Deductive reasoning by itself can never tell us about facts. The use of deduction in science is to serve as a calculus to make our observations go further, not to take the place of observation." (Arthur D Ritchie, "Scientific Method: An Inquiry into the Character and Validity of Natural Laws", 1923)

"Every philosophy is tinged with the coloring of some secret imaginative, which never emerges explicitly into its trains of reasoning." (Alfred N Whitehead, "Science and the Modern World", 1925)

"At first sight the assimilative tendency shown by thought seems sufficient to secure stability in judgments. To assimilate, in psychology as in biology, is to reproduce oneself by means of the external world; it is to transform perceptions until they are identical with one’s own thought, i.e. with previous schemas. Assimilation is therefore preservation and, in a certain sense, identification." (Jean Piaget, "Judgement and Reasoning in the Child", 1928)

"In the presence of certain objects of thought or of certain affirmations the child, in virtue of previous experiences, adopts a certain way of reacting and thinking which is always the same, and which might be called a schema of reasoning. Such schemas are the functional equivalents of general propositions, but since the child is not conscious of these schemas before discussion and a desire for proof have laid them bare and at the same time changed their character, they cannot be said to constitute implicit general propositions. They simply constitute certain unconscious tendencies which live their own life but are submitted to no general systematization and consequently lead to no logical exactitude. To put it in another way, they form a logic of action but not yet a logic of thought." (Jean Piaget, "Judgement and Reasoning in the Child", 1928)

🪷On Mind: On Reasoning (1910-1919)

 "It is doubtful if we have any other subject that does so much to bring to the front the danger of carelessness, of slovenly reasoning, of inaccuracy, and of forgetfulness as this science of geometry, which has been so polished and perfected as the centuries have gone on." (David E Smith, "The Teaching of Geometry", 1911)

"The theoretical side of physical chemistry is and will probably remain the dominant one; it is by this peculiarity that it has exerted such a great influence upon the neighboring sciences, pure and applied, and on this ground physical chemistry may be regarded as an excellent school of exact reasoning for all students of the natural sciences." (Svante Arrhenius, "Theories of Solutions", 1912) 

"[…] mathematical verities flow from a small number of self-evident propositions by a chain of impeccable reasonings; they impose themselves not only on us, but on nature itself. They fetter, so to speak, the Creator and only permit him to choose between some relatively few solutions. A few experiments then will suffice to let us know what choice he has made. From each experiment a number of consequences will follow by a series of mathematical deductions, and in this way each of them will reveal to us a corner of the universe. This, to the minds of most people, and to students who are getting their first ideas of physics, is the origin of certainty in science." (Henri Poincaré, "The Foundations of Science", 1913)

"And here is what makes this analysis situs interesting to us; it is that geometric intuition really intervenes there. When, in a theorem of metric geometry, one appeals to this intuition, it is because it is impossible to study the metric properties of a figure as abstractions of its qualitative properties, that is, of those which are the proper business of analysis situs. It has often been said that geometry is the art of reasoning correctly from badly drawn figures. This is not a capricious statement; it is a truth that merits reflection. But what is a badly drawn figure? It is what might be executed by the unskilled draftsman spoken of earlier; he alters the properties more or less grossly; his straight lines have disquieting zigzags; his circles show awkward bumps. But this does not matter; this will by no means bother the geometer; this will not prevent him from reasoning." (Henri Poincaré, "Dernières pensées", 1913)

"Philosophy, like science, consists of theories or insights arrived at as a result of systemic reflection or reasoning in regard to the data of experience. It involves, therefore, the analysis of experience and the synthesis of the results of analysis into a comprehensive or unitary conception. Philosophy seeks a totality and harmony of reasoned insight into the nature and meaning of all the principal aspects of reality." (Joseph A Leighton, "The Field of Philosophy: An outline of lectures on introduction to philosophy," 1919)

"Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise." (David Hilbert,"Natur und Mathematisches Erkennen", 1919–20)

🪷On Mind: On Reasoning (1900-1909)

"[...] it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning." (David Hilbert, "Mathematical Problems", 1900) 

"We form in the imagination some sort of diagrammatic, that is, iconic, representation of the facts, as skeletonized as possible. The impression of the present writer is that with ordinary persons this is always a visual image, or mixed visual and muscular; but this is an opinion not founded on any systematic examination." (Charles S Peirce, "Notes on Ampliative Reasoning", 1901)

"Does the mathematical method proceed from particular to the general, and, if so, how can it be called deductive? […] If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from a syllogism." (Henri Poincaré, "Science and Hypothesis", 1901)

"No theorem can be new unless a new axiom intervenes in its demonstration; reasoning can only give us immediately evident truths borrowed from direct intuition; it would only be an intermediary parasite." (Henri Poincaré, "Science and Hypothesis", 1901)

"Syllogistic reasoning remains incapable of adding anything to the data that are given it; the data are reduced to axioms, and that is all we should find in the conclusions." (Henri Poincaré, "Science and Hypothesis", 1901)

"The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinite number of syllogisms. […] We see, then, that in reasoning by recurrence we confine ourselves to the enunciation of the minor of the first syllogism, and the general formula which contains as particular cases all the majors. This unending series of syllogisms is thus reduced to a phrase of a few lines." (Henri Poincaré, "Science and Hypothesis", 1901)

"To undertake the calculation of any probability, and even for that calculation to have any meaning at all, we must admit, as a point of departure, an hypothesis or convention which has always something arbitrary about it. In the choice of this convention we can be guided only by the principle of sufficient reason. Unfortunately, this principle is very vague and very elastic, and in the cursory examination we have just made we have seen it assume different forms. The form under which we meet it most often is the belief in continuity, a belief which it would be difficult to justify by apodeictic reasoning, but without which all science would be impossible. Finally, the problems to which the calculus of probabilities may be applied with profit are those in which the result is independent of the hypothesis made at the outset, provided only that this hypothesis satisfies the condition of continuity." (Henri Poincaré, "Science and Hypothesis", 1901)

"Treatises on mechanics do not clearly distinguish between what is experiment, what is mathematical reasoning, what is convention, and what is hypothesis." (Henri Poincaré, "Science and Hypothesis", 1901)

"We form in the imagination some sort of diagrammatic, that is, iconic, representation of the facts, as skeletonized as possible. The impression of the present writer is that with ordinary persons this is always a visual image, or mixed visual and muscular; but this is an opinion not founded on any systematic examination." (Charles S Peirce, "Notes on Ampliative Reasoning", 1901)

"By no process of sound reasoning can a conclusion drawn from limited data have more than a limited application." (Jospeph W Mellor, "Higher Mathematics for Students of Chemistry and Physics", 1902)

"We imagine cases, place mental diagrams before our mind's eye, and multiply these cases, until a habit is formed of expecting that always to turn out the case, which has been seen to be the result in all the diagrams. To appeal to such a habit is a very different thing from appealing to any immediate instinct of rationality. That the process of forming a habit of reasoning by the use of diagrams is often performed there is no room for doubt. It is perfectly open to consciousness." (Charles S Peirce, "Fallibility of Reasoning and the Feeling of Rationality", cca. 1902)

"An axiom is a self-evident truth, the statement of which is superfluous to the conclusiveness of the reasoning, and which only serves to show a principle involved in the reasoning. It is generally a truth of observation; such as the assertion that something is true." (Charles S Peirce, "New Elements" ["Kaina stoiceia"], 1904)

"We believe that in our reasonings we no longer appeal to intuition; the philosophers will tell us this is an illusion. Pure logic could never lead us to anything but tautologies; it could create nothing new; not from it alone can any science issue. In one sense these philosophers are right; to make arithmetic, as to make geometry, or to make any science, something else than pure logic is necessary. To designate this something else we have no word other than intuition. But how many different ideas are hidden under this same word?" (Henri Poincaré, "Intuition and Logic in Mathematics", 1905)

"Diagrammatic reasoning is the only really fertile reasoning. If logicians would only embrace this method, we should no longer see attempts to base their science on the fragile foundations of metaphysics or a psychology not based on logical theory; and there would soon be such an advance in logic that every science would feel the benefit of it." (Charles S Peirce, "Prolegomena to an Apology for Pragmaticism", Monist 16(4), 1906)

"The requisites for the axioms are various. They should be simple, in the sense that each axiom should enumerate one and only one statement. The total number of axioms should be few. A set of axioms must be consistent, that is to say, it must not be possible to deduce the contradictory of any axiom from the other axioms. According to the logical 'Law of Contradiction,' a set of entities cannot satisfy inconsistent axioms. Thus the existence theorem for a set of axioms proves their consistency. Seemingly this is the only possible method of proof of consistency. "Alfred N Whitehead, "The axioms of projective geometry, 1906) 

"Theory is not solely an economical representation of experimental laws; it is also a classification of these laws. […] theory, by developing the numerous ramifications of the deductive reasoning which connects principles to experimental laws, establishes an order and a classification among these laws. It brings some laws together, closely arranged in the same group; it separates some of the others by placing them in two groups very far apart. Theory gives, so to speak, the table of contents and the chapter headings under which the science to be studied will be methodically divided, and it indicates the laws which are to be arranged under each of these chapters." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)

"The type of reasoning found in mathematics seems thus not only available but essentially interwoven with every inference in non-mathematical reasoning, being always used in one of its two steps ; facility in making the other step, the more difficult one, must be attained through other than purely mathematical training." (Jacob W A Young, "The Teaching of Mathematics", 1907)

“Mathematics is merely a shorthand method of recording physical intuition and physical reasoning, but it should not be a formalism leading from nowhere to nowhere, as it is likely to be made by one who does not realize its purpose as a tool." (Charles P Steinmetz,"Transactions of the American Institute of Electrical Engineers", 1909)

“We should always aim toward the economy of thought. It is not enough to give models for imitation. It must be possible to pass beyond these models and, in place of repeating their reasoning at length each time, to sum this in a few words." (Jules H Poincaré, 1909)

🪷On Mind: On Reasoning (1875-1899)

"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

"[…] 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)

"[…] 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)

"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) 

"There is probably no other science which presents such different appearances to one who cultivates it and one who does not, as mathematics. To [the noncultivator] it is ancient, venerable, and complete; a body of dry, irrefutable, unambiguous reasoning. To the mathematician, on the other hand, his science is yet in the purple bloom of vigorous youth, everywhere stretching out after the 'attainable but unattained', and full of the excitement of nascent thoughts; its logic is beset with ambiguities, and its analytic processes, like Bunyan's road, have a quagmire on one side and a deep ditch on the other, and branch off into innumerable by-paths that end in a wilderness." (Charles H Chapman, Bulletin of the New York Mathematical Society 2, 1892) 

"[…] geometry is the art of reasoning well from badly drawn figures; however, these figures, if they are not to deceive us, must satisfy certain conditions; the proportions may be grossly altered, but the relative positions of the different parts must not be upset." (Henri Poincaré, 1895)

"Deduction is that mode of reasoning which examines the state of things asserted in the premises, forms a diagram of that state of things, perceives in the parts of the diagram relations not explicitly mentioned in the premises, satisfies itself by mental experiments upon the diagram that these relations would always subsist, or at least would do so in a certain proportion of cases, and concludes their necessary, or probable, truth." (Charles S Peirce,"Kinds of Reasoning", cca. 1896)

"A system is a set of objects compromising all that stands to one another in a group of connected relations." (Charles S Peirce, "Cambridge Lectures on Reasoning and the Logic of Things: Detached Ideas on Vitally Important Topics", 1898) 

"The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thoughts, or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus." (Alfred N Whitehead,"Universal Algebra", 1898) 

"The whole of Mathematics consists in the organization of a series of aids to the imagination in the process of reasoning." (Alfred North Whitehead,"A Treatise on Universal Algebra", 1898)

"Many theorems are obvious upon looking at a moderately-sized figure; but the reasoning must be such as to convince the mind of their truth when, from excessive increase or diminution of the scale, the figures themselves have past the boundary even of imagination." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)

"The ordinary logic has a great deal to say about genera and species, or in our nineteenth century dialect, about classes. Now a class is a set of objects compromising all that stand to one another in a special relation of similarity. But where ordinary logic talks of classes the logic of relatives talks of systems. A system is a set of objects compromising all that stands to one another in a group of connected relations. Induction according to ordinary logic rises from the contemplation of a sample of a class to that of a whole class; but according to the logic of relatives it rises from the contemplation of a fragment of a system to the envisagement of the complete system." (Charles S Peirce, "Cambridge Lectures on Reasoning and the Logic of Things: Detached Ideas on Vitally Important Topics", 1898) 

"Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner. In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that. If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum." (Henri Poincaré, 1899)

🪷On Mind: On Reasoning (1850-1874)

"The reasoning of mathematicians is founded on certain and infallible principles. Every word they use conveys a determinate idea, and by accurate definitions they excite the same ideas in the mind of the reader that were in the mind of the writer. When they have defined the terms they intend to make use of, they premise a few axioms, or self-evident principles, that every one must assent to as soon as proposed. They then take for granted certain postulates, that no one can deny them, such as, that a right line may be drawn from any given point to another, and from these plain, simple principles they have raised most astonishing speculations, and proved the extent of the human mind to be more spacious and capacious than any other science." (John Adams,"Diary", 1850)

"The discovery of truth is prevented more effectively, not by the false appearance things present and which mislead into error, not directly by weakness of the reasoning powers, but by preconceived opinion, by prejudice." (Arthur Schopenhauer, "Parerga and Paralipomena", 1851)

"Reasoning from analogy is often most plausible and most deceptive." (Charles Simmons, "A Laconic Manual and Brief Remarker", 1852)

"To deduce the laws of the symbols of Logic from a consideration of those operations of the mind which are implied in the strict use of language as an instrument of reasoning." (George Boole, "An Investigation of the Laws of Thought", 1854)

 "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)

"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)