On Logic (-1799)

"Among all the liberal arts, the first is logic, and specifically that part of logic which gives initial instruction about words. [...] the word 'logic' has a broad meaning, and is not restricted exclusively to the science of argumentative reasoning. [It includes] Grammar [which] is 'the science of speaking and writing correctly - the starting point of all liberal studies'." (John of Salisbury, "Metalogicon", 1159)

"Among the various branches of philosophy, logic has two prerogatives: it has both the honor of coming first and the distinction of serving as an efficacious instrument throughout the whole body [of philosophy]. Natural and moral philosophers can construct their principles only by the forms of proof supplied by logicians. Also, in order to define and divide correctly, they must borrow and employ the art of the logicians. And if, perchance, they succeed in this without logic, their success is due to luck, rather than to science. Logic is 'rational' [philosophy], and we may readily see from the very name, what progress in philosophy can be expected from one who [since he lacks logic] lacks reason." (John of Salisbury, "Metalogicon", 1159)

"The method of demonstration is therefore generally feeble and ineffective with regard to facts of nature (I refer to corporeal and changeable things). But it quickly recovers its strength when applied to the field of mathematics. For whatever it concludes in regard to such things as numbers, proportions and figures is indubitably true, and cannot be otherwise. One who wishes to become a master of the science of demonstration should first obtain a good grasp of probabilities. Whereas the principles of demonstrative logic are necessary; those of dialectic are probable." (John of Salisbury, "Metalogicon", 1159)

"[…] he who wishes to attain to human perfection, must therefore first study Logic, next the various branches of Mathematics in their proper order, then Physics, and lastly Metaphysics." (Moses Maimonides, "The Guide for the Perplexed", 1190)

"Men are further beholding […] generally to chance, or anything else, than to logic, for the invention or arts and sciences." (Francis Bacon, "The Advancement of Learning", 1605)

"XI. As the present sciences are useless for the discovery of effects, so the present system of logic is useless for the discovery of the sciences. XII. The present system of logic rather assists in confirming and rendering inveterate the errors founded on vulgar notions than in searching after truth, and is therefore more hurtful than useful."  (Sir Francis Bacon, "Novum Organum", 1620)

"In logic, they teach that contraries laid together more evidently appear: it follows, then, that all controversy being permitted, falsehood will appear more false, and truth the more true; which must needs conduce much to the general confirmation of an implicit truth." (John Milton, "True Religion, Heresy, Schism, Toleration, and what best means may be used against the Growth of Popery", 1673)

"Nothing universal can be rationally affirmed on any moral or any political subject. Pure metaphysical abstraction does not belong to these matters. The lines of morality are not like the ideal lines of mathematics. They are broad and deep as well as long. They admit of exceptions; they demand modifications. These exceptions and modifications are not made by the process of logic, but by the rules of prudence. Prudence is not only the first in rank of the virtues political and moral, but she is the director, regulator, the standard of them all. Metaphysics cannot live without definition; but prudence is cautious how she defines." (Edmund Burke, "Appeal from the New to the Old Whigs", 1791)

"The advantages which mathematics derives from the peculiar nature of those relations about which it is conversant, from its simple and definite phraseology, and from the severe logic so admirably displayed in the concatenation of its innumerable theorems, are indeed immense, and well entitled to separate and ample illustration." (Dugald Stewart, "Philosophy of the Human Mind", 1792) 

On Logic (1800-1849)

"Whether or not I have found a logic, by the role of which operations with imaginary quantities are conducted, is not now the question. but surely this is evident that since they lead to right conclusions they must have a logic! […] Till the doctrines of negative and imaginary quantities are better taught than they are at present taught in the University of Cambridge, I agree with you that they had better not be taught [...]" (Robert Woodhouse, [letter to Baron Meseres] 1801)

"Logic and metaphysics make use of more tools than all the rest of the sciences put together, and do the least work." (Charles C Colton, "Remarks on the Talents of Lord Byron and the Tendencies of Don Juan" 1823)

"Geometry, then, is the application of strict logic to those properties of space and figure which are self-evident, and which therefore cannot be disputed. But the rigor of this science is carried one step further; for no property, however evident it may be, is allowed to pass without demonstration, if that can be given. The question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1830)

"It is not easy to anatomize the constitution and the operations of a mind which makes such an advance in knowledge. Yet we may observe that there must exist in it, in an eminent degree, the elements which compose the mathematical talent. It must possess distinctness of intuition, tenacity and facility in tracing logical connection, fertility of invention, and a strong tendency to generalization." (William Whewell, "History of the Inductive Sciences" Vol. 1, 1837)

"Logicians may reason about abstractions. But the great mass of men must have images." (Thomas B Macaulay, Critical and Miscellaneous Essays, 1840)

"Logic is the procession or proportionate unfolding of the intuition; but its virtue is as silent method; the moment it would appear as propositions, and have a separate value, it is worthless." (Ralph W Emerson, "Essays", 1841)

"Logic does not pretend to teach the surgeon what are the symptoms which indicate a violent death. This he must learn from his own experience and observation, or from that of others, his predecessors in his peculiar science. But logic sits in judgment on the sufficiency of that observation and experience to justify his rules, and on the sufficiency of his rules to justify his conduct. It does not give him proofs, but teaches him what makes them proofs, and how he is to judge of them." (John Stuart Mill, "A System of Logic, Ratiocinative and Inductive: Being a Connected View of the Principles of Evidence, and the Methods of Scientific Investigation", 1843)

On Logic (1850-1874)

"The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore, the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man's mind." (James C Maxwell, 1850)

"[Algebra] has for its object the resolution of equations; taking this expression in its full logical meaning, which signifies the transformation of implicit functions into equivalent explicit ones. In the same way arithmetic may be defined as destined to the determination of the values of functions. […] We will briefly say that Algebra is the Calculus of functions, and Arithmetic is the Calculus of Values." (Auguste Comte, "Philosophy of Mathematics", 1851)

"It is, after all, a principle of logic not to multiply entities unnecessarily." (Antoine-Laurent Lavoisier, "Réflexions sur le phlogistique", 1862)

"There is a kind, I might almost say, of artistic satisfaction, when we are able to survey the enormous wealth of Nature as a regularly ordered whole - a kosmos, an image of the logical thought of our own mind." (Hermann von Helmholtz. "On the Conservation of Force", 1862)

"If an idea presents itself to us, we must not reject it simply because it does not agree with the logical deductions of a reigning theory." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"The purely formal sciences, logic and mathematics, deal with such relations which are independent of the definite content, or the substance of the objects, or at least can be. In particular, mathematics involves those relations of objects to each other that involve the concept of size, measure, number." (Hermann Hankel, "Theorie der Complexen Zahlensysteme", 1867)

"I think it would be desirable that this form of word [mathematics] should be reserved for the applications of the science, and that we should use mathematic in the singular to denote the science itself, in the same way as we speak of logic, rhetoric, or (own sister to algebra) music." (James J Sylvester, [Presidential Address to the British Association] 1869)

"A law of nature, however, is not a mere logical conception that we have adopted as a kind of memoria technical to enable us to more readily remember facts. We of the present day have already sufficient insight to know that the laws of nature are not things which we can evolve by any speculative method. On the contrary, we have to discover them in the facts; we have to test them by repeated observation or experiment, in constantly new cases, under ever-varying circumstances; and in proportion only as they hold good under a constantly increasing change of conditions, in a constantly increasing number of cases with greater delicacy in the means of observation, does our confidence in their trustworthiness rise." (Hermann von Helmholtz, "Popular Lectures on Scientific Subjects", 1873)

"So intimate is the union between Mathematics and Physics that probably by far the larger part of the accessions to our mathematical knowledge have been obtained by the efforts of mathematicians to solve the problems set to them by experiment, and to create for each successive class phenomena a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature. Sometimes the mathematician has been before the physicist, and it has happened that when some great and new question has occurred to the experimentalist or the observer, he has found in the armory of the mathematician the weapons which he needed ready made to his hand. But much oftener, the questions proposed by the physicist have transcended the utmost powers of the mathematics of the time, and a fresh mathematical creation has been needed to supply the logical instrument requisite to interpret the new enigma." (Henry J S Smith, Nature, Volume 8, 1873)

"The invention of a new symbol is a step in the advancement of civilisation. Why were the Greeks, in spite of their penetrating intelligence and their passionate pursuit of Science, unable to carry Mathematics farther than they did? and why, having formed the conception of the Method of Exhaustions, did they stop short of that of the Differential Calculus? It was because they had not the requisite symbols as means of expression. They had no Algebra. Nor was the place of this supplied by any other symbolical language sufficiently general and flexible; so that they were without the logical instruments necessary to construct the great instrument of the Calculus." (George H Lewes "Problems of Life and Mind", 1873)

"The rules of Arithmetic operate in Algebra; the logical operations supposed to be peculiar to Ideation operate in Sensation, There is but one Calculus, but one Logic; though for convenience we divide the one into Arithmetic the calculus of values, and Algebra the calculus of relations; the other into the Logic of Feeling and the Logic of Signs." (George H Lewes "Problems of Life and Mind", 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 from the values assigned." (George H Lewes "Problems of Life and Mind", 1873)

On Logic (1875-1899)

"Some definite interpretation of a linear algebra would, at first sight, appear indispensable to its successful application. But on the contrary, it is a singular fact, and one quite consonant with the principles of sound logic, that its first and general use is mostly to be expected from its want of significance. The interpretation is a trammel to the use. Symbols are essential to comprehensive argument." (Benjamin Peirce, "On the Uses and Transformations of Linear Algebra", 1875)

"The most striking characteristic of the written language of algebra and of the higher forms of the calculus is the sharpness of definition, by which we are enabled to reason upon the symbols by the mere laws of verbal logic, discharging our minds entirely of the meaning of the symbols, until we have reached a stage of the process where we desire to interpret our results. The ability to attend to the symbols, and to perform the verbal, visible changes in the position of them permitted by the logical rules of the science, without allowing the mind to be perplexed with the meaning of the symbols until the result is reached which you wish to interpret, is a fundamental part of what is called analytical power. Many students find themselves perplexed by a perpetual attempt to interpret not only the result, but each step of the process. They thus lose much of the benefit of the labor-saving machinery of the calculus and are, indeed, frequently incapacitated for using it." (Thomas Hill, "Uses of Mathesis", Bibliotheca Sacra Vol. 32 (127), 1875)

"I say that a manifold (a collection, a set) of elements that belong to any conceptual sphere is well-defined, when on the basis of its definition and as a consequence of the logical principle of excluded middle it must be regarded as internally determined, both whether an object pertaining to the same conceptual sphere belongs or not as an element to the manifold, and whether two objects belonging to the set are equal to each other or not, despite formal differences in the ways of determination." (Georg Cantor, "Ober unendliche, lineare Punktmannichfaltigkeiten", 1879)

"[…] 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)

"We need a system of symbols from which every ambiguity is banned, which has a strict logical form from which the content cannot escape." (Gottlob Frege, "Über die wissenschaftliche berechtigung einer begriffsschrift", Zeitschrift für Philosophie und philosophische Kritik 81, 1882)

"Logic works, metaphysics contemplates." (Joseph Joubert, "The Notebooks of Joseph Joubert", 1883)

"The steps to scientific as well as other knowledge consist in a series of logical fictions which are as legitimate as they are indispensable in the operations of thought, but whose relations to the phenomena whereof they are the partial and not unfrequently merely symbolical representations must never be lost sight of." (John Stallo, "The Concepts and Theories of Modern Physics", 1884)

"The unimaginability of the content of a word is no reason, then, to deny it any meaning or to exclude it from usage. That we are nevertheless inclined to do so is probably owing to the fact that we consider words individually and ask about their meaning [in isolation], for which we then adopt a mental picture. Thus a word for which we are lacking a corresponding inner picture will seem to have no content. However, we must always consider a complete sentence. Only in [the context of] the latter do the words really have a meaning. The inner pictures which somehow sway before us (in reading the sentence) need not correspond to the logical components of the judgment. It is enough if the sentence as a whole has a sense; by means of this its parts also receive their content." (Gottlob Frege, "The Foundations of Arithmetic" , 1884)

"In calling arithmetic (algebra, analysis) just a part of logic, I declare already that I take the number-concept to be completely independent of the ideas or intuitions of space and time, that I see it as an immediate product of the pure laws of thought." (Richard Dedekind, "Was sind und was sollen die Zahlen?", 1888)

"In science nothing capable of proof ought to be accepted without proof. Though this demand seems so reasonable yet I cannot regard it as having been met even in […] that part of logic which deals with the theory of numbers. In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number concept entirely independent of the notions of intuition of space and time, that I consider it an immediate result from the laws of thought." (Richard Dedekind, "Was sind und was sollen die Zahlen?", 1888)

"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)

" […] the naive intuition is not exact, while the refined intuition is not properly intuition at all, but arises through the logical development from axioms considered as perfectly exact." (Felix Klein, [lectures] 1893)

"At the basis of our Symbolic Logic, however represented, whether by words by letters or by diagrams, we shall always find the same state of things. What we ultimately have to do is to break up the entire field before us into a definite number of classes or compartments which are mutually exclusive and collectively exhaustive." (John Venn, "Symbolic Logic" 2nd Ed., 1894)

"It is not easy to anatomize the constitution and the operations of a mind which makes such an advance in knowledge. Yet we may observe that there must exist in it, in an eminent degree, the elements which compose the mathematical talent. It must possess distinctness of intuition, tenacity and facility in tracing logical connection, fertility of invention, and a strong tendency to generalization." (William Whewell, "History of the Inductive Sciences" Vol. 1, 1894)

"The images which we may form of things are not determined without ambiguity by the requirement that the consequents of the images must be the images of the consequents. Various images of the same objects are possible, and these images may differ in various respects. We should at once denote as inadmissible all images which implicitly contradict the laws of our thought. Hence we postulate in the first place that all our images shall be logically permissible or, briefly, that they shall be permissible. We shall denote as incorrect any permissible images, if their essential relations contradict the relations of external things, i.e. if they do not satisfy our first fundamental requirement. Hence we postulate in the second place that our images shall be correct." (Heinrich Hertz, "The Principles of Mechanics Presented in a New Form", 1894)

"Even one well-made observation will be enough in many cases, just as one well-constructed experiment often suffices for the establishment of a law." (Émile Durkheim, "The Rules of Sociological Method", "The Rules of Sociological Method", 1895)

"It is they who hold the secret of the mysterious property of the mind by which error ministers to truth, and truth slowly but irrevocably prevails. Theirs is the logic of discovery, the demonstration of the advance of knowledge and the development of ideas, which as the earthly wants and passions of men remain almost unchanged, are the charter of progress, and the vital spark in history." (Lord John Acton, "The Study of History", [lecture delivered at Cambridge] 1895)

"[In mathematics] we behold the conscious logical activity of the human mind in its purest and most perfect form. Here we learn to realize the laborious nature of the process, the great care with which it must proceed, the accuracy which is necessary to determine the exact extent of the general propositions arrived at, the difficulty of forming and comprehending abstract concepts; but here we learn also to place confidence in the certainty, scope and fruitfulness of such intellectual activity." (Hermann Helmholtz, "Vorträge und Reden", 1896)

"In mathematics we see the conscious logical activity of our mind in its purest and most perfect form; here is made manifest to us all the labor and the great care with which it progresses, the precision which is necessary to determine exactly the source of the established general theorems, and the difficulty with which we form and comprehend abstract conceptions; but we also learn here to have confidence in the certainty, breadth, and fruitfulness of such intellectual labor." (Hermann von Helmholtz, "Vorträge und Reden", 1896)

"Incidentally, naive intuition, which is in large part an inherited talent, emerges unconsciously from the in-depth study of this or that field of science. The word ‘Anschauung’ has not perhaps been suitably chosen. I would like to include here the motoric sensation with which an engineer assesses the distribution of forces in something he is designing, and even that vague feeling possessed by the experienced number cruncher about the convergence of infinite processes with which he is confronted. I am saying that, in its fields of application, mathematical intuition understood in this way rushes ahead of logical thinking and in each moment has a wider scope than the latter " (Felix Klein, "Über Arithmetisierung der Mathematik", Zeitschrift für mathematischen und naturwissen-schaftlichen Unterricht 27, 1896)

"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)

On Logic (2010-2019)

 "[…] a conceptual model is a diagram connecting variables and constructs based on theory and logic that displays the hypotheses to be tested." (Mary Wolfinbarger Celsi et al, "Essentials of Business Research Methods", 2011)

"A proof in logic and mathematics is, traditionally, a deductive argument from some given assumptions to a conclusion. Proofs are meant to present conclusive evidence in the sense that the truth of the conclusion should follow necessarily from the truth of the assumptions. Proofs must be, in principle, communicable in every detail, so that their correctness can be checked." (Sara Negri  & Jan von Plato, "Proof Analysis", 2011)

"Like classical logic, fuzzy logic uses formulas to formally represent statements about the world. Given an appropriate semantic structure (such as an evaluation of propositional symbols in the case of propositional logic, or a relational structure in the case of predicate logic), a truth degree of formula ? is denoted by ||?||. It is significant that the truth degree ||?|| of ? may in general be any element of the set of truth degrees. That is, formulas in fuzzy logic are true to degrees , not just true or false as in the case of classical logic." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)

"Nevertheless, the use of fuzzy logic is supported by at least the following three arguments. First, fuzzy logic is rooted in the intuitively appealing idea that the truths of propositions used by humans are a matter of degree. An important consequence is that the basic principles and concepts of fuzzy logic are easily understood. Second, fuzzy logic has led to many successful applications, including many commercial products, in which the crucial part relies on representing and dealing with statements in natural language that involve vague terms. Third, fuzzy logic is a proper generalization of classical logic, follows an agenda similar to that of classical logic, and has already been highly developed. An important consequence is that fuzzy logic extends the rich realm of applications of classical logic to applications in which the bivalent character of classical logic is a limiting factor." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)

"The principal idea employed by fuzzy logic is to allow for a partially ordered scale of truth-values, called also truth degrees, which contains the values representing false and true , but also some additional, intermediary truth degrees. That is, the set {0,1} of truth-values of classical logic, where 0 and 1 represent false and true , respectively, is replaced in fuzzy logic by a partially ordered scale of truth degrees with the smallest degree being 0 and the largest one being 1." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)

"There is no way to guarantee in advance what pure mathematics will later find application. We can only let the process of curiosity and abstraction take place, let mathematicians obsessively take results to their logical extremes, leaving relevance far behind, and wait to see which topics turn out to be extremely useful. If not, when the challenges of the future arrive, we won’t have the right piece of seemingly pointless mathematics to hand." (Peter Rowlett, "The Unplanned Impact of Mathematics", Nature Vol. 475 (7355), 2011)

"We use the term fuzzy logic to refer to all aspects of representing and manipulating knowledge that employ intermediary truth-values. This general, commonsense meaning of the term fuzzy logic encompasses, in particular, fuzzy sets, fuzzy relations, and formal deductive systems that admit intermediary truth-values, as well as the various methods based on them." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)

"Complex systems defy intuitive solutions. Even a third-order, linear differential equation is unsolvable by inspection. Yet, important situations in management, economics, medicine, and social behavior usually lose reality if simplified to less than fifth-order nonlinear dynamic systems. Attempts to deal with nonlinear dynamic systems using ordinary processes of description and debate lead to internal inconsistencies. Underlying assumptions may have been left unclear and contradictory, and mental models are often logically incomplete. Resulting behavior is likely to be contrary to that implied by the assumptions being made about' underlying system structure and governing policies." (Jay W Forrester, "Modeling for What Purpose?", The Systems Thinker Vol. 24 (2), 2013)

"Mathematical intuition is the mind’s ability to sense form and structure, to detect patterns that we cannot consciously perceive. Intuition lacks the crystal clarity of conscious logic, but it makes up for that by drawing attention to things we would never have consciously considered." (Ian Stewart, "Visions of Infinity", 2013)

"Proof, in fact, is the requirement that makes great problems problematic. Anyone moderately competent can carry out a few calculations, spot an apparent pattern, and distil its essence into a pithy statement. Mathematicians demand more evidence than that: they insist on a complete, logically impeccable proof. Or, if the answer turns out to be negative, a disproof. It isn’t really possible to appreciate the seductive allure of a great problem without appreciating the vital role of proof in the mathematical enterprise. Anyone can make an educated guess. What’s hard is to prove it’s right. Or wrong." (Ian Stewart, "Visions of Infinity", 2013)

"Abstraction is an essential knowledge process, the process (or, to some, the alleged process) by which we form concepts. It consists in recognizing one or several common features or attributes (properties, predicates) in individ­uals, and on that basis stating a concept subsuming those common features or attributes. Concept is an idea, associated with a word expressing a prop­erty or a collection of properties inferred or derived from different samples. Subsumption is the logical technique to get generality from particulars." (Hourya B Sinaceur," Facets and Levels of Mathematical Abstraction", Standards of Rigor in Mathematical Practice 18-1, 2014)

"Mathematical abstraction is the process of considering and manipulating op­erations, rules, methods and concepts divested from their reference to real world phenomena and circumstances, and also deprived from the content con­nected to particular applications. […] abstraction is the process of passing from things to ideas, properties and relations, to properties of relations and relations of properties, to properties of relations between properties, etc. Being a fundamental thinking process, abstraction has two faces: a logical face and evidently a psychological aspect that is the target of cognitive sciences." (Hourya B Sinaceur,"Facets and Levels of Mathematical Abstraction", Standards of Rigor in Mathematical Practice 18-1, 2014)

"A conceptual model is a framework that is initially used in research to outline the possible courses of action or to present an idea or thought. When a conceptual model is developed in a logical manner, it will provide a rigor to the research process." (N Elangovan & R Rajendran, "Conceptual Model: A Framework for Institutionalizing the Vigor in Business Research", 2015)

"One of the most powerful transformational catalysts is knowledge, new information, or logic that defies old mental models and ways of thinking. […] The key to transforming mental models is to interrupt the automatic responses that are driven by the old model and respond differently based on the new model. Each time you are able to do this, you are actually loosening the old circuit and creating new neural connections in your brain, often referred to as self-directed neuroplasticity." (Elizabeth Thornton, "The Objective Leader", 2015)

"Science, at its core, is simply a method of practical logic that tests hypotheses against experience. Scientism, by contrast, is the worldview and value system that insists that the questions the scientific method can answer are the most important questions human beings can ask, and that the picture of the world yielded by science is a better approximation to reality than any other." (John M Greer, "After Progress: Reason and Religion at the End of the Industrial Age", 2015)

"Cybernetics is an interdisciplinary science. It originated ‘at the junction’ of mathematics, logic, semiotics, physiology, biology and sociology. Among its inherent features, we mention analysis and revelation of general principles and approaches in scientific cognition. Control theory, communication theory, operations research and others represent most weighty theories within cybernetics 1.0." (Dmitry A Novikov, "Cybernetics 2.0", 2016)

"Mathematical rigour is the thing that enables mathematicians to agree with one another about what is and isn’t correct, rather than just having arguments about competing theories and never coming to a conclusion. Mathematics is based on the rules of logic, the idea being that if you only use objects that behave strictly according to the rules of logic, then as long as you only strictly apply the rules of logic, no disagreements can ever arise."(Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Mathematics starts with the process of stripping away the ambiguities and leaving only things that can be unambiguously manipulated according to logic. It continues by then manipulating those things according to logic to see what happens." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

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

On Logic (2000-2009)

"Simple observation generally gets us nowhere. It is the creative imagination that increases our understanding by finding connections between apparently unrelated phenomena, and forming logical, consistent theories to explain them. And if a theory turns out to be wrong, as many do, all is not lost. The struggle to create an imaginative, correct picture of reality frequently tells us where to go next, even when science has temporarily followed the wrong path." (Richard Morris, "The Universe, the Eleventh Dimension, and Everything: What We Know and How We Know It", 1999)

"Mathematics is not placid, static and eternal. […] Most mathematicians are happy to make use of those axioms in their proofs, although others do not, exploring instead so-called intuitionist logic or constructivist mathematics. Mathematics is not a single monolithic structure of absolute truth!" (Gregory J Chaitin, "A century of controversy over the foundations of mathematics", 2000)

"The seeming absence of any ascertained organizing principle in the distribution or the succession of the primes had bedeviled mathematicians for centuries and given Number Theory much of its fascination. Here was a great mystery indeed, worthy of the most exalted intelligence: since the primes are the building blocks of the integers and the integers the basis of our logical understanding of the cosmos, how is it possible that their form is not determined by law? Why isn't 'divine geometry' apparent in their case?" (Apostolos Doxiadis, "Uncle Petros and Goldbach's Conjecture", 2000)

"What does a rigorous proof consist of? The word ‘proof’ has a different meaning in different intellectual pursuits. A ‘proof’ in biology might consist of experimental data confirming a certain hypothesis; a ‘proof’ in sociology or psychology might consist of the results of a survey. What is common to all forms of proof is that they are arguments that convince experienced practitioners of the given field. So too for mathematical proofs. Such proofs are, ultimately, convincing arguments that show that the desired conclusions follow logically from the given hypotheses." (Ethan Bloch, "Proofs and Fundamentals", 2000)

"Zero is behind all of the big puzzles in physics. The infinite density of the black hole is a division by zero. The big bang creation from the void is a division by zero. The infinite energy of the vacuum is a division by zero. Yet dividing by zero destroys the fabric of mathematics and the framework of logic - and threatens to undermine the very basis of science. […] The universe begins and ends with zero." (Charles Seife ."Zero, the Biography of a Dangerous Idea", 2000)

"[…] interval mathematics and fuzzy logic together can provide a promising alternative to mathematical modeling for many physical systems that are too vague or too complicated to be described by simple and crisp mathematical formulas or equations. When interval mathematics and fuzzy logic are employed, the interval of confidence and the fuzzy membership functions are used as approximation measures, leading to the so-called fuzzy systems modeling." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"That a proof must be convincing is part of the anthropology of mathematics, providing the key to understanding mathematics as a human activity. We invoke the logic of mathematics when we demand that every informal proof be capable of being formalized within the confines of a definite formal system. Finally, the epistemology of mathematics comes into play with the requirement that a proof be surveyable. We can't really say that we have created a genuine piece of knowledge unless it can be examined and verified by others; there are no private truths in mathematics." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"[…] we underestimate the share of randomness in about everything […]  The degree of resistance to randomness in one’s life is an abstract idea, part of its logic counterintuitive, and, to confuse matters, its realizations nonobservable." (Nassim N Taleb, "Fooled by Randomness", 2001)

"The fuzzy set theory is taking the same logical approach as what people have been doing with the classical set theory: in the classical set theory, as soon as the two-valued characteristic function has been defined and adopted, rigorous mathematics follows; in the fuzzy set case, as soon as a multi-valued characteristic function (the membership function) has been chosen and fixed, a rigorous mathematical theory can be fully developed." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"While mathematical truth is the aim of inquiry, some falsehoods seem to realize this aim better than others; some truths better realize the aim than other truths and perhaps even some falsehoods realize the aim better than some truths do. The dichotomy of the class of propositions into truths and falsehoods should thus be supplemented with a more fine-grained ordering - one which classifies propositions according to their closeness to the truth, their degree of truth-likeness or verisimilitude. The problem of truth-likeness is to give an adequate account of the concept and to explore its logical properties and its applications to epistemology and methodology." (Graham Oddie, "Truth-likeness", Stanford Encyclopedia of Philosophy, 2001)

"One can be highly functionally numerate without being a mathematician or a quantitative analyst. It is not the mathematical manipulation of numbers (or symbols representing numbers) that is central to the notion of numeracy. Rather, it is the ability to draw correct meaning from a logical argument couched in numbers. When such a logical argument relates to events in our uncertain real world, the element of uncertainty makes it, in fact, a statistical argument." (Eric R Sowey, "The Getting of Wisdom: Educating Statisticians to Enhance Their Clients' Numeracy", The American Statistician 57(2), 2003)

"Pure mathematics was characterized by an obsession with proof, rigor, beauty, and elegance, and sought its foundations in the disembodied worlds of logic or intuition. Far from being coextensive with physics, pure mathematics could be ‘applied’ only after it had been made foundationally secure by the purists." (Andrew Warwick, "Masters of Theory: Cambridge and the rise of mathematical physics", 2003)

"We start from vague pictures or ideas […] which we encapsulate by rules, and then we discover that those rules persuade us to modify our mental images. We engage in a dialog between our mental images and our ability to justify them via equations. As we understand what we are investigating more clearly, the pictures become sharper and the equations more elaborate. Only at the end of the process does anything like a formal set of axioms followed by logical proofs" (E Brian Davies, "Science in the Looking Glass", 2003)

"A theorem is never arrived at in the way that logical thought would lead you to believe or that posterity thinks. It is usually much more accidental, some chance discovery in answer to some kind of question. Eventually you can rationalize it and say that this is how it fits. Discoveries never happen as neatly as that. You can rewrite history and make it look much more logical, but actually it happens quite differently." (Sir Michael Atiyah, 2004)

"Nature does weird things. It lives on the edge. But it is careful to bob and weave from the fatal punch of logical paradox." (Brian Greene, "The Fabric of the Cosmos: Space, Time, and the Texture of Reality", 2004)

"The inner mysteries of quantum mechanics require a willingness to extend one’s mental processes into a strange world of phantom possibilities, endlessly branching into more and more abstruse chains of coupled logical networks, endlessly extending themselves forward and even backwards in time." (John C Ward, "Memoirs of a Theoretical Physicist", 2004)

"It seems that scientists are often attracted to beautiful theories in the way that insects are attracted to flowers - not by logical deduction, but by something like a sense of smell." (Steven Weinberg, "Physics Today", 2005)

"Mathematics is not about abstract entities alone but is about relation of abstract entities with real entities. […] Adequacy relations between abstract and real entities provide space or opportunity where mathematical and logical thought operates parsimoniously."  (Navjyoti Singh, "Classical Indian Mathematical Thought", 2005)

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

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

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

"There is a big debate as to whether logic is part of mathematics or mathematics is part of logic. We use logic to think. We notice that our thinking, when it is valid, goes in certain patterns. These patterns can be studied mathematically. Thus, logic is a part of mathematics, called 'mathematical logic'." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006) 

"[…] statistical thinking, though powerful, is never as easy or automatic as simply plugging numbers into formulas. In order to use statistical methods appropriately, you need to understand their logic, not just the computing rules." (Ann E Watkins et al, "Statistics in Action: Understanding a World of Data", 2007)

"Human language is a vehicle of truth but also of error, deception, and nonsense. Its use, as in the present discussion, thus requires great prudence. One can improve the precision of language by explicit definition of the terms used. But this approach has its limitations: the definition of one term involves other terms, which should in turn be defined, and so on. Mathematics has found a way out of this infinite regression: it bypasses the use of definitions by postulating some logical relations (called axioms) between otherwise undefined mathematical terms. Using the mathematical terms introduced with the axioms, one can then define new terms and proceed to build mathematical theories. Mathematics need, not, in principle rely on a human language. It can use, instead, a formal presentation in which the validity of a deduction can be checked mechanically and without risk of error or deception." (David Ruelle, "The Mathematician's Brain", 2007)

"It thus stands to reason that mathematical structures have a dual origin: in part human, in part purely logical. Human mathematics requires short formulations (because of our poor memory, etc.). But mathematical logic dictates that theorems with a short formulation may have very long proofs, as shown by Gödel. Clearly you don't want to go through the same long proof again and again. You will try instead to use repeatedly the short theorem that you have obtained. And an important tool to obtain short formulations is to give short names to mathematical objects that occur repeatedly. These short names describe new concepts. So we see how concept creation arises in the practice of mathematics as a consequence of the inherent logic of the sub ject and of the nature of human mathematicians." (David Ruelle, "The Mathematician's Brain", 2007)

"Logic moves in one direction, the direction of clarity, coherence, and structure. Ambiguity moves in the other direction, that of fluidity, openness, and release. Mathematics moves back and forth between these two poles. Mathematics is not a fixed, static entity that can be structured definitively. It is dynamic, alive: its dynamism a function of the relationship between the two poles that have been described above. It is the interactions between these different aspects that give mathematics its power." (William Byers, "How Mathematicians Think", 2007)

"Mathematics as done by mathematicians is not just heaping up statements logically deduced from the axioms. Most such statements are rubbish, even if perfectly correct. A good mathematician will look for interesting results. These interesting results, or theorems, organize themselves into meaningful and natural structures, and one may say that the object of mathematics is to find and study these structures." (David Ruelle, "The Mathematician's Brain", 2007)

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

"We speak of mathematical reality as we speak of physical reality. They are different but both quite real. Mathematical reality is of logical nature, while physical reality is tied to the universe in which we live and which we perceive through our senses. This is not to say that we can readily define mathematical or physical reality, but we can relate to them by making mathematical proofs or physical experiments." (David Ruelle, "The Mathematician's Brain", 2007)

"With mathematics and particularly mathematical logic, we come to grips with the most remote, the most nonhuman objects that the human mind has encountered. And this icy remoteness exerts on some people an irresistible fascination." (David Ruelle, "The Mathematician's Brain", 2007)

"A worldview must be coherent, logical and adequate. Coherence means that the fundamental ideas constituting the worldview must be seen as proceeding from a single, unifying, overarching concept. A logical worldview means simply that the various ideas constituting it should not be contradictory. Adequate means that it is capable of explaining, logically and coherently, every element of contemporary experience." (M. G. Jackson, "Transformative Learning for a New Worldview: Learning to Think Differently", 2008)

"As students, we learned mathematics from textbooks. In textbooks, mathematics is presented in a rigorous and logical way: definition, theorem, proof, example. But it is not discovered that way. It took many years for a mathematical subject to be understood well enough that a cohesive textbook could be written. Mathematics is created through slow, incremental progress, large leaps, missteps, corrections, and connections." (Richard S Richeson, "Eulers Gem: The Polyhedron Formula and the birth of Topology", 2008)

"Therefore, mathematical ecology does not deal directly with natural objects. Instead, it deals with the mathematical objects and operations we offer as analogs of nature and natural processes. These mathematical models do not contain all information about nature that we may know, but only what we think are the most pertinent for the problem at hand. In mathematical modeling, we have abstracted nature into simpler form so that we have some chance of understanding it. Mathematical ecology helps us understand the logic of our thinking about nature to help us avoid making plausible arguments that may not be true or only true under certain restrictions. It helps us avoid wishful thinking about how we would like nature to be in favor of rigorous thinking about how nature might actually work." (John Pastor, "Mathematical Ecology of Populations and Ecosystems", 2008)

"Zero is the mathematically defined numerical function of nothingness. It is used not for an evasion but for an apprehension of reality. Zero is by far the most interesting number among all the others: It is a symbol for what is not there. It is an emptiness that increases any number it's added to. Zero is an inexhaustible and indispensable paradox. Zero is the only number which can be divided by every other number. Zero is also only number which can divide no other number. It seems zero is also the most debated number in mathematics. We know that mathematicians are involved in heated philosophical and logical discussions around the issues of zero: Can we divide a number by zero? Is the result of this division infinity or not? Is zero a positive or a negative number? Is it even or is it odd?" (Fahri Karakas, "Reflections on zero and zero-centered spirituality in organizations", 2008)

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

On Logic (1990-1999)

"On this view, we recognize science to be the search for algorithmic compressions. We list sequences of observed data. We try to formulate algorithms that compactly represent the information content of those sequences. Then we test the correctness of our hypothetical abbreviations by using them to predict the next terms in the string. These predictions can then be compared with the future direction of the data sequence. Without the development of algorithmic compressions of data all science would be replaced by mindless stamp collecting - the indiscriminate accumulation of every available fact. Science is predicated upon the belief that the Universe is algorithmically compressible and the modern search for a Theory of Everything is the ultimate expression of that belief, a belief that there is an abbreviated representation of the logic behind the Universe's properties that can be written down in finite form by human beings." (John D Barrow, New Theories of Everything", 1991)

"The cybernetics phase of cognitive science produced an amazing array of concrete results, in addition to its long-term (often underground) influence: the use of mathematical logic to understand the operation of the nervous system; the invention of information processing machines (as digital computers), thus laying the basis for artificial intelligence; the establishment of the metadiscipline of system theory, which has had an imprint in many branches of science, such as engineering (systems analysis, control theory), biology (regulatory physiology, ecology), social sciences (family therapy, structural anthropology, management, urban studies), and economics (game theory); information theory as a statistical theory of signal and communication channels; the first examples of self-organizing systems. This list is impressive: we tend to consider many of these notions and tools an integrative part of our life […]" (Francisco Varela, "The Embodied Mind", 1991)

"The most persuasive positive argument for mental images as objects is [that] whenever one thinks one is seeing something there must be something one is seeing. It might be an object directly, or it might be a mental picture. [This] point is so plausible that it is deniable only at the peril of becoming arbitrary. One should concede that the question whether mental images are entities of some sort is not resolvable by logical or linguistic analysis, and believe what makes sense of experience." (Eva T H Brann,"The World of Imagination" , 1991)

"This absolutist view of mathematical knowledge is based on two types of assumptions: those of mathematics, concerning the assumption of axioms and definitions, and those of logic concerning the assumption of axioms, rules of inference and the formal language and its syntax. These are local or micro-assumptions. There is also the possibility of global or macro-assumptions, such as whether logical deduction suffices to establish all mathematical truths." (Paul Ernest, "The Philosophy of Mathematics Education", 1991)

"[…] mathematics is not just an austere, logical structure of forbidding purity, but also a vital, vibrant instrument for understanding the world, including the workings of our minds, and this aspect of mathematics was all but lost." (Mark Kac,  "Mathematics: Tensions", 1992)

"Mathematical modeling is about rules - the rules of reality. What distinguishes a mathematical model from, say, a poem, a song, a portrait or any other kind of ‘model’, is that the mathematical model is an image or picture of reality painted with logical symbols instead of with words, sounds or watercolors." (John L Casti, "Reality Rules, The Fundamentals", 1992)

"Pedantry and sectarianism aside, the aim of theoretical physics is to construct mathematical models such as to enable us, from the use of knowledge gathered in a few observations, to predict by logical processes the outcomes in many other circumstances. Any logically sound theory satisfying this condition is a good theory, whether or not it be derived from ‘ultimate’ or ‘fundamental’ truth." (Clifford Truesdell & Walter Noll, "The Non-Linear Field Theories of Mechanics" 2nd Ed., 1992)

"The popular image of mathematics as a collection of precise facts, linked together by well-defined logical paths, is revealed to be false. There is randomness and hence uncertainty in mathematics, just as there is in physics." (Paul Davis, "The Mind of God", 1992)

"The binary logic of modern computers often falls short when describing the vagueness of the real world. Fuzzy logic offers more graceful alternatives." (Bart Kosko & Satoru Isaka, "Fuzzy Logic,” Scientific American Vol. 269, 1993)

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

"The insight at the root of artificial intelligence was that these 'bits' (manipulated by computers) could just as well stand as symbols for concepts that the machine would combine by the strict rules of logic or the looser associations of psychology." (Daniel Crevier, "AI: The tumultuous history of the search for artificial intelligence", 1993)

"The word theory, as used in the natural sciences, doesn’t mean an idea tentatively held for purposes of argument - that we call a hypothesis. Rather, a theory is a set of logically consistent abstract principles that explain a body of concrete facts. It is the logical connections among the principles and the facts that characterize a theory as truth. No one element of a theory [...] can be changed without creating a logical contradiction that invalidates the entire system. Thus, although it may not be possible to substantiate directly a particular principle in the theory, the principle is validated by the consistency of the entire logical structure." (Alan Cromer, "Uncommon Sense: The Heretical Nature of Science", 1993)

"But our ways of learning about the world are strongly influenced by the social preconceptions and biased modes of thinking that each scientist must apply to any problem. The stereotype of a fully rational and objective ‘scientific method’, with individual scientists as logical (and interchangeable) robots, is self-serving mythology." (Stephen J Gould, "This View of Life: In the Mind of the Beholder", "Natural History", Vol. 103, No. 2, 1994)

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

"People have amazing facilities for sensing something without knowing where it comes from (intuition); for sensing that some phenomenon or situation or object is like something else (association); and for building and testing connections and comparisons, holding two things in mind at the same time (metaphor). These facilities are quite important for mathematics. Personally, I put a lot of effort into ‘listening’ to my intuitions and associations, and building them into metaphors and connections. This involves a kind of simultaneous quieting and focusing of my mind. Words, logic, and detailed pictures rattling around can inhibit intuitions and associations." (William P Thurston, "On proof and progress in mathematics", Bulletin of the American Mathematical Society Vol. 30 (2), 1994)

"The sequence for the understanding of mathematics may be: intuition, trial, error, speculation, conjecture, proof. The mixture and the sequence of these events differ widely in different domains, but there is general agreement that the end product is rigorous proof – which we know and can recognize, without the formal advice of the logicians. […] Intuition is glorious, but the heaven of mathematics requires much more. Physics has provided mathematics with many fine suggestions and new initiatives, but mathematics does not need to copy the style of experimental physics. Mathematics rests on proof - and proof is eternal." (Saunders Mac Lan, "Reponses to …",m Bulletin of the American Mathematical Society Vol. 30 (2), 1994)

"An intuitive proof allows you to understand why the theorem must be true; the logic merely provides firm grounds to show that it is true." (Ian Stewart, "Concepts of Modern Mathematics",  1995)

"In contemplating natural phenomena, we frequently have to distinguish between effective complexity and logical depth. For example, the apparently complicated pattern of energy levels of atomic nuclei might easily be misattributed to some complex law at the fundamental level, but it is now believed to follow from a simple underlying theory of quarks, gluons, and photons, although lengthy calculations would be required to deduce the detailed pattern from the basic equations. Thus the pattern has a good deal of logical depth and very little effective complexity." (Murray Gell-Mann, "What is Complexity?", Complexity Vol. 1 (1), 1995)

"Music and higher mathematics share some obvious kinship. The practice of both requires a lengthy apprenticeship, talent, and no small amount of grace. Both seem to spring from some mysterious workings of the mind. Logic and system are essential for both, and yet each can reach a height of creativity beyond the merely mechanical." (Frederick Pratter, "How Music and Math Seek Truth in Beauty", Christian Science Monitor, 1995)

"Probability theory is an ideal tool for formalizing uncertainty in situations where class frequencies are known or where evidence is based on outcomes of a sufficiently long series of independent random experiments. Possibility theory, on the other hand, is ideal for formalizing incomplete information expressed in terms of fuzzy propositions." (George Klir, "Fuzzy sets and fuzzy logic", 1995)

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

"Networks constitute the new social morphology of our societies, and the diffusion of networking logic substantially modifies the operation and outcomes in processes of production, experience, power, and culture. While the networking form of social organization has existed in other times and spaces, the new information technology paradigm provides the material basis for its pervasive expansion throughout the entire social structure." (Manuel Castells, "The Rise of the Network Society", 1996)

"So we pour in data from the past to fuel the decision-making mechanisms created by our models, be they linear or nonlinear. But therein lies the logician's trap: past data from real life constitute a sequence of events rather than a set of independent observations, which is what the laws of probability demand.[...] It is in those outliers and imperfections that the wildness lurks." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)

"The logic of the emotional mind is associative; it takes elements that symbolize a reality, or trigger a memory of it, to be the same as that reality. That is why similes, metaphors and images speak directly to the emotional mind." (Daniel Goleman, "Emotional Intelligence", 1996)

"Math has its own inherent logic, its own internal truth. Its beauty lies in its ability to distill the essence of truth without the messy interference of the real world. It’s clean, neat, above it all. It lives in an ideal universe built on the geometer’s perfect circles and polygons, the number theorist’s perfect sets. It matters not that these objects don’t exist in the real world. They are articles of faith." (K C Cole, "The Universe and the Teacup: The Mathematics of Truth and Beauty", 1997)

"Mathematical logic deals not with the truth but only with the game of truth." (Gian-Carlo Rota, "Indiscrete Thoughts", 1997)

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

"A formal system consists of a number of tokens or symbols, like pieces in a game. These symbols can be combined into patterns by means of a set of rules which defines what is or is not permissible (e.g. the rules of chess). These rules are strictly formal, i.e. they conform to a precise logic. The configuration of the symbols at any specific moment constitutes a ‘state’ of the system. A specific state will activate the applicable rules which then transform the system from one state to another. If the set of rules governing the behaviour of the system are exact and complete, one could test whether various possible states of the system are or are not permissible." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

"Mathematical truth is found to exceed the proving of theorems and to elude total capture in the confining meshes of any logical net." (John Polkinghorne, "Belief in God in an Age of Science", 1998)

"As systems became more varied and more complex, we find that no single methodology suffices to deal with them. This is particularly true of what may be called information intelligent systems - systems which form the core of modern technology. To conceive, design, analyze and use such systems we frequently have to employ the totality of tools that are available. Among such tools are the techniques centered on fuzzy logic, neurocomputing, evolutionary computing, probabilistic computing and related methodologies. It is this conclusion that formed the genesis of the concept of soft computing." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic: A personal perspective", 1999)

On Logic (1980-1989)

"My suggestion is that at each state the proper order of operation of the mind requires an overall grasp of what is generally known, not only in formal logical, mathematical terms, but also intuitively, in images, feelings, poetic usage of language, etc." (David Bohm,"Wholeness and the Implicate Order Wholeness and the Implicate Order", 1980)

"Science attempts to find logic and simplicity in nature. Mathematics attempts to establish order and simplicity in human thought." (Edward Teller, "The Pursuit of Simplicity", 1980)

"Some people believe that a theorem is proved when a logically correct proof is given; but some people believe a theorem is proved only when the student sees why it is inevitably true." (Wesley R Hamming, "Coding and Information Theory", 1980)

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

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

"A proof in science does more than eliminate doubt. It eliminates inconsistencies and provides the underlying logical basis of the statement." (Edward Teller, "The Pursuit of Simplicity", 1981)

"Moreover, ‘fact’ doesn’t mean ‘absolute certainty’; there ain’t no such animal in an exciting and complex world. The final proofs of logic and mathematics flow deductively from stated premises and achieve certainty only because they are NOT about the empirical world. Evolutionists make no claim for perpetual truth, though creationists often do (and then attack us falsely for a style of argument that they themselves favor)." (Stephen J Gould, "Evolution as Fact and Theory", Discover, 1981)

"Philosophical objections may be raised by the logical implications of building a mathematical structure on the premise of fuzziness, since it seems (at least superficially) necessary to require that an object be or not be an element of a given set. From an aesthetic viewpoint, this may be the most satisfactory state of affairs, but to the extent that mathematical structures are used to model physical actualities, it is often an unrealistic requirement. [...] Fuzzy sets have an intuitively plausible philosophical basis. Once this is accepted, analytical and practical considerations concerning fuzzy sets are in most respects quite orthodox." (James Bezdek, 1981)

"Heavy dependence on direct observation is essential to biology not only because of the complexity of biological phenomena, but because of the intervention of natural selection with its criterion of adequacy rather than perfection. In a system shaped by natural selection it is inevitable that logic will lose its way." (George A Bartholomew, "Scientific innovation and creativity: a zoologist’s point of view", American Zoologist Vol. 22, 1982)

"The purpose of scientific enquiry is not to compile an inventory of factual information, nor to build up a totalitarian world picture of Natural Laws in which every event that is not compulsory is forbidden. We should think of it rather as a logically articulated structure of justifiable beliefs about nature. It begins as a story about a Possible World - a story which we invent and criticize and modify as we go along, so that it winds by being, as nearly as we can make it, a story about real life." (Sir Peter B Medawar, "Pluto’s Republic", 1982)

"[…] a mathematician's ultimate concern is that his or her inventions be logical, not realistic. This is not to say, however, that mathematical inventions do not correspond to real things. They do, in most, and possibly all, cases. The coincidence between mathematical ideas and natural reality is so extensive and well documented, in fact, that it requires an explanation. Keep in mind that the coincidence is not the outcome of mathematicians trying to be realistic - quite to the contrary, their ideas are often very abstract and do not initially appear to have any correspondence to the real world. Typically, however, mathematical ideas are eventually successfully applied to describe real phenomena […] "(Michael Guillen,"Bridges to Infinity: The Human Side of Mathematics", 1983)

"[…] mathematics is not a science – it is not capable of proving or disproving the existence of real things. In fact, a mathematician’s ultimate concern is that his or her inventions be logical, not realistic." (Michael Guillen, "Bridges to Infinity: The Human Side of Mathematics", 1983)

"The theory of the visual display of quantitative information consists of principles that generate design options and that guide choices among options. The principles should not be applied rigidly or in a peevish spirit; they are not logically or mathematically certain; and it is better to violate any principle than to place graceless or inelegant marks on paper. Most principles of design should be greeted with some skepticism, for word authority can dominate our vision, and we may come to see only though the lenses of word authority rather than with our own eyes." (Edward R Tufte, "The Visual Display of Quantitative Information", 1983)

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

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

"The nothingness ‘before’ the creation of the universe is the most complete void that we can imagine - no space, time, or matter existed. It is a world without place, without duration or eternity, without number - it is what mathematicians call ‘the empty set’. Yet this unthinkable void converts itself into the plenum of existence - a necessary consequence of physical laws. Where are these laws written into that void? What ‘tells’ the void that is pregnant with a possible universe? It would seem that, even the void is subject to law, a logic that exists prior to space and time." (Heinz R Pagels, "Perfect Symmetry: The Search for the Beginning of Time", 1985)

"Mathematics is good if it enriches the subject, if it opens up new vistas, if it solves old problems, if it fills gaps, fitting snugly and satisfyingly into what is already known, or if it forges new links between previously unconnected parts of the subject It is bad if it is trivial, overelaborate, or lacks any definable mathematical purpose or direction It is pure if its methods are pure - that is, if it doesn't cheat and tackle one problem while pretending to tackle another, and if there are no gaping holes in its logic It is applied if it leads to useful insights outside mathematics By these criteria, today's mathematics contains as high a proportion of good work as at any other period, and as any other area, and much of it manages to be both pure and applied at the same time." (Ian Stewart, "The Problems of Mathematics", 1987)

"There are elements of freedom in mathematics. We can decide in favor of one thing or another. Reference to the permanence principle (or another principle) is not a logical argument. We are free to opt for one or another. But we are not free when it comes to the consequences. We achieve harmony if we opt for a certain one (that minus times minus is plus). By making this choice we make the same choice as others in the past and present." (Ernst Schuberth, "Minus mal Minus", Forum Pädagogik, Vol. 2, 1988)

"As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than observation as its standard of truth, yet employs observation, simulation, and even experimentation as a means of discovering truth. "(National Research Council, "Everybody Counts", 1989)

"People might suppose that a mathematical proof is conceived as a logical progression, where each step follows upon the ones that have preceded it. Yet the conception of a new argument is hardly likely actually to proceed in this way. There is a globality and seemingly vague conceptual content that is necessary in the construction of a mathematical argument; and this can bear little relation to the time that it would seem to take in order fully to appreciate a serially presented proof" (Roger Penrose, "The Emperor’s New Mind", 1989)

"To function in today's society, mathematical literacy - what the British call ‘numeracy' - is as essential as verbal literacy […] Numeracy requires more than just familiarity with numbers. To cope confidently with the demands of today's society, one must be able to grasp the implications of many mathematical concepts - for example, change, logic, and graphs - that permeate daily news and routine decisions - mathematical, scientific, and cultural - provide a common fabric of communication indispensable for modern civilized society. Mathematical literacy is especially crucial because mathematics is the language of science and technology." (National Research Council, "Everybody counts: A report to the nation on the future of mathematics education", 1989)

On Logic (1970-1979)

"In general, one might define a complex of semantic components connected by logical constants as a concept. The dictionary of a language is then a system of concepts in which a phonological form and certain syntactic and morphological characteristics are assigned to each concept. This system of concepts is structured by several types of relations. It is supplemented, furthermore, by redundancy or implicational rules […] representing general properties of the whole system of concepts. […] At least a relevant part of these general rules is not bound to particular languages, but represents presumably universal structures of natural languages. They are not learned, but are rather a part of the human ability to acquire an arbitrary natural language." (Manfred Bierwisch, "Semantics", 1970)

"Engineers, as a rule are not and do not pretend to be philosophers in the sense of building up consistent systems of thought following logically from certain premises. If anything, they pride themselves on being hard-headed practical men concerned only with facts, disdaining mere speculation or opinion. In practice, however, engineers do make many assumptions about the nature of the universe, of man, and of society." (Edwin T Layton Jr., "The Revolt of the Engineers", 1971)

"Physics is not a finished logical system. Rather, at any moment it spans a great confusion of ideas, some that survive like folk epics from the heroic periods of the past, and others that arise like utopian novels from our dim premonitions of a future grand synthesis." (Steven Weinberg, "Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity", 1972)

"The result of the implementation, the logical design, is traditionally shown as a series of block diagrams. These blocks represent in effect a series of statements, Actually, a direct presentation of these statements is more suitable and, although less familiar, more easily understood. The Harvard Mark IV was to large degree designed and described by such statements, as has been the case with several subsequent developments." (Gerrit Blaauw, "Computer Architecture", 1972)

"There are two subcategories of holist called irredundant holists and redundant holists. Students of both types image an entire system of facts or principles. Though an irredundant holist's image is rightly interconnected, it contains only relevant and essential constitents. In contrast, redundant holists entertain images that contain logically irrelevant or overspecific material, commonly derived from data used to 'enrich' the curriculum, and these students embed the salient facts and principles in a network of redundant items. Though logically irrelevant, the items in question are of great psychological importance to a 'redundant holist', since he uses them to access, retain and manipulate whatever he was originally required to learn." (Gordon Pask, "Learning Strategies and Individual Competence", 1972)

"There is little point in demanding minor concessions and relaxations of the abstract, timeless general equilibrium. The light it can throw on human affairs is throw by its most austere and formal version. We are not concerned to ask: How could it possibly work? The useful question is: What does its logical structure imply?" (George L S Shackle, "Epistemics and Economics", 1972)

"This parallel, between cybernetic explanation and the tactics of logical or mathematical proof, is of more than trivial interest. Outside of cybernetics, we look for explanation, but not for anything which would simulate logical proof. This simulation of proof is something new. We can say, however, with hindsight wisdom, that explanation by simulation of logical or mathematical proof was expectable. After all, the subject matter of cybernetics is not events and objects but the information 'carried' by events and objects. We consider the objects or events only as proposing facts, propositions, messages, percepts, and the like. The subject matter being propositional, it is expectable that explanation would simulate the logical." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"And yet, on looking into the history of science, one is overwhelmed by evidence that all too often there is no regular procedure, no logical system of discovery, no simple, continuous development. The process of discovery has been as varied as the temperament of the scientist." (Gerald Holton, "Thematic Origins of Scientific Thought: Kepler to Einstein", 1973)

"[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)

"An intuitive proof allows you to understand why the theorem must be true; the logic merely provides firm grounds to show that it is true." (Ian Stewart, "Concepts of Modern Mathematics", 1975)

"A model […] is a story with a specified structure: to explain this catch phrase is to explain what a model is. The structure is given by the logical and mathematical form of a set of postulates, the assumptions of the model. The structure forms an uninterpreted system, in much the way the postulates of a pure geometry are now commonly regarded as doing. The theorems that follow from the postulates tell us things about the structure that may not be apparent from an examination of the postulates alone." (Allan Gibbard & Hal R. Varian, "Economic Models", The Journal of Philosophy, Vol. 75, No. 11, 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)

"Common to both logical positivism and transformational linguistics is their view of language-as-mathematics. Both focus on language as a system of primitive or elementary units which can be combined according to fixed rules. However useful this analogy may be in certain limited ways, it creates problems in understanding how the purely formal system of elements and rules relates to something other than itself. Both create dualistic systems which oppose formal linguistic competence to empirical components." (Stephen A Tyler, "The said and the unsaid: Mind, meaning, and culture", 1978)

"On the face of it there should be no disagreement about mathematical proof. Everybody looks enviously at the alleged unanimity of mathematicians; but in fact there is a considerable amount of controversy in mathematics. Pure mathematicians disown the proofs of applied mathematicians, while logicians in turn disavow those of pure mathematicians. Logicists disdain the proofs of formalists and some intuitionists dismiss with contempt the proofs of logicists and formalists." (Imre Lakatos,"Mathematics, Science and Epistemology" Vol. 2, 1978)

"[...] despite an objectivity about mathematical results that has no parallel in the world of art, the motivation and standards of creative mathematics are more like those of art than of science. Aesthetic judgments transcend both logic and applicability in the ranking of mathematical theorems: beauty and elegance have more to do with the value of a mathematical idea than does either strict truth or possible utility." (Lynn A Steen, "Mathematics Today: Twelve Informal Essays", Mathematics Today, 1978)

"[…] a body of practices widely regarded by outsiders as well organized, logical, and coherent, in fact consists of a disordered array of observations with which scientists struggle to produce order." (Bruno Latour & S Woolgar, Laboratory Life: The Social Construction of Scientific Facts, 1979)

"In set theory, perhaps more than in any other branch of mathematics, it is vital to set up a collection of symbolic abbreviations for various logical concepts. Because the basic assumptions of set theory are absolutely minimal, all but the most trivial assertions about sets tend to be logically complex, and a good system of abbreviations helps to make otherwise complex statements."  (Keith Devlin, "Sets, Functions, and Logic: An Introduction to Abstract Mathematics", 1979)

On Logic (1960-1969)

"It is inherent in the logical character of the abstract self-organizing system that all available methods of organization are used, and that it cannot be realized in a single reference frame. Thus, any of the tricks which the physical model can perform, such as learning and remembering, may be performed by one or all of a variety of mechanisms, chemical or electrical or mechanical." (Gordon Pask, "The Natural History of Networks", 1960)

"Analogy is even slipperier than logic." (Robert A Heinlein, "Stranger in a Strange Land", 1961)

"Model-making, the imaginative and logical steps which precede the experiment, may be judged the most valuable part of scientific method because skill and insight in these matters are rare. Without them we do not know what experiment to do. But it is the experiment which provides the raw material for scientific theory. Scientific theory cannot be built directly from the conclusions of conceptual models." (Herbert G Andrewartha, "Introduction to the Study of Animal Population", 1961) 

"Although the drama of games of strategy is strongly linked with the psychological aspects of the conflict, game theory is not concerned with these aspects. Game theory, so to speak, plays the board. It is concerned only with the logical aspects of strategy." (Anatol Rapoport, "The Use and Misuse of Game Theory", Scientific American 207, 1962)

"Cybernetics is concerned primarily with the construction of theories and models in science, without making a hard and fast distinction between the physical and the biological sciences. The theories and models occur both in symbols and in hardware, and by 'hardware’ we shall mean a machine or computer built in terms of physical or chemical, or indeed any handleable parts. Most usually we shall think of hardware as meaning electronic parts such as valves and relays. Cybernetics insists, also, on a further and rather special condition that distinguishes it from ordinary scientific theorizing: it demands a certain standard of effectiveness. In this respect it has acquired some of the same motive power that has driven research on modern logic, and this is especially true in the construction and application of artificial languages and the use of operational definitions. Always the search is for precision and effectiveness, and we must now discuss the question of effectiveness in some detail. It should be noted that when we talk in these terms we are giving pride of place to the theory of automata at the expense, at least to some extent, of feedback and information theory." (Frank H George, "The Brain As A Computer", 1962)

"Living mathematics rests on the fluctuation between the antithesis powers of intuition and logic, the individuality of 'grounded' problems and the generality of far-reaching abstractions. We ourselves must prevent the development being forced to only one pole of the life-giving antithesis." (Richard Courant, 1962)

one thing. In bringing techniques of logical and mathematical analysis gives men an opportunity to bring conflicts up from the level of fights, where the intellect is beclouded by passions, to the level of games, where the intellect has a chance to operate." (Anatol Rapoport, "The Use and Misuse of Game Theory", Scientific American 207, 1962)

"Science is a matter of disinterested observation, patient ratiocination within some system of logically correlated concepts. In real-life conflicts between reason and passion the issue is uncertain. Passion and prejudice are always able to mobilize their forces more rapidly and press the attack with greater fury; but in the long run (and often, of course, too late) enlightened self-interest may rouse itself, launch a counterattack and win the day for reason." (Aldous L Huxley, "Literature and Science", 1963)

"With even a superficial knowledge of mathematics, it is easy to recognize certain characteristic features: its abstractions, its precision, its logical rigor, the indisputable character of its conclusions, and finally, the exceptionally broad range for its applications." (Aleksandr D Aleksandrov, 1963)

"Mathematicians create by acts of insight and intuition. Logic then sanctions the conquests of intuition. It is the hygiene that mathematics practices to keep its ideas healthy and strong. Moreover, the whole structure rests fundamentally on uncertain ground, the intuition of humans. Here and there an intuition is scooped out and replaced by a firmly built pillar of thought; however, this pillar is based on some deeper, perhaps less clearly defined, intuition. Though the process of replacing intuitions with precise thoughts does not change the nature of the ground on which mathematics ultimately rests, it does add strength and height to the structure." (Morris Kline, "Mathematics in Western Culture ", 1964)

"[Game theory is] essentially a structural theory. It uncovers the logical structure of a great variety of conflict situations and describes this structure in mathematical terms. Sometimes the logical structure of a conflict situation admits rational decisions; sometimes it does not." (Anatol Rapoport, "Prisoner's dilemma: A study in conflict and cooperation", 1965)

"As soon as we inquire into the reasons for the phenomena, we enter the domain of theory, which connects the observed phenomena and traces them back to a single ‘pure’ phenomena, thus bringing about a logical arrangement of an enormous amount of observational material." (Georg Joos, "Theoretical Physics", 1968)

"A theorem is no more proved by logic and computation than a sonnet is written by grammar and rhetoric, or than a sonata is composed by harmony and counterpoint, or a picture painted by balance and perspective." (George Spencer-Brown, "Laws of Form", 1969)

On Logic (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)

"Perhaps the greatest paradox of all is that there are paradoxes in mathematics […] because mathematics builds on the old but does not discard it, because its theorems are deduced from postulates by the methods of logic, in spite of its having undergone revolutionary changes we do not suspect it of being a discipline capable of engendering paradoxes." (James R Newman, "Mathematics and the Imagination", 1940)

"Perhaps the greatest paradox of all is that there are paradoxes in mathematics […] because mathematics builds on the old but does not discard it, because its theorems are deduced from postulates by the methods of logic, in spite of its having undergone revolutionary changes we do not suspect it of being a discipline capable of engendering paradoxes." (James R Newman, "Mathematics and the Imagination", 1940)

"Science is the attempt to make the chaotic diversity of our sense experience correspond to a logically uniform system of thought." (Albert Einstein, "Considerations Concerning the Fundaments of Theoretical Physics", Science Vol. 91 (2369), 1940)

"The revolution in scientific ideas just mentioned is primarily logical. It is due to recognition that the very method of physical science, with its primary standard units of mass, space, and time, is concerned with measurements of relations of change, not with individuals as such." (John Dewey, "Time and Individuality", 1940)

"Mathematicians deal with possible worlds, with an infinite number of logically consistent systems. Observers explore the one particular world we inhabit. Between the two stands the theorist. He studies possible worlds but only those which are compatible with the information furnished by observers. In other words, theory attempts to segregate the minimum number of possible worlds which must include the actual world we inhabit. Then the observer, with new factual information, attempts to reduce the list further. And so it goes, observation and theory advancing together toward the common goal of science, knowledge of the structure and observation of the universe." (Edwin P Hubble, "The Problem of the Expanding Universe", 1941)

"Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science." (Richard Courant & Herbert Robbins, "What Is Mathematics?", 1941)

"The fact that the proof of a theorem consists in the application of certain simple rules of logic does not dispose of the creative element in mathematics, which lies in the choice of the possibilities to be examined." (Richard Courant & Herbert Robbins, "What Is Mathematics?: An Elementary Approach to Ideas and Methods", 1941)

"To say that mathematics in general has been reduced to logic hints at some new firming up of mathematics at its foundations. This is misleading. Set theory is less settled and more conjectural than the classical mathematical superstructure than can be founded upon it." (Willard van Orman Quine, "Elementary Logic", 1941)

"The faith of scientists in the power of mathematics is so implicit that their work has gradually become less and less observation, and more and more calculation. The promiscuous collection and tabulation of data have given way to a process of assigning possible meanings, merely supposed real entities, to mathematical terms, working out the logical results, and then staging certain crucial experiments to check the hypothesis against the actual, empirical results. But the facts [...] accepted by virtue of these tests are not actually observed at all." (Susanne K Langer, "Philosophy in a New Key", 1942)

"Since we consider purposefulness a concept necessary for the understanding of certain modes of behavior we suggest that a teleological study is useful if it avoids problems of causality and concerns itself merely with an investigation of purpose." (Arturo Rosenblueth, Norbert Wiener & Julian Bigelow, "Behavior, Purpose and Technology", Philosophy of Science Vol. 10 (1), 1943)

"A material model is the representation of a complex system by a system which is assumed simpler and which is also assumed to have some properties similar to those selected for study in the original complex system. A formal model is a symbolic assertion in logical terms of an idealised relatively simple situation sharing the structural properties of the original factual system." (Arturo Rosenblueth & Norbert Wiener, "The Role of Models in Science", Philosophy of Science Vol. 12 (4), 1945)

"A mathematician is not a man who can readily manipulate figures; often he cannot. He is not even a man who can readily perform the transformations of equations by the use of calculus. He is primarily an individual who is skilled in the use of symbolic logic on a high plane, and especially he is a man of intuitive judgment in the choice of the manipulative processes he employs." (Vannevar Bush, "As We May Think", 1945)

"Every time one combines and records facts in accordance with established logical processes, the creative aspect of thinking is concerned only with the selection of the data and the process to be employed, and the manipulation thereafter is repetitive in nature and hence a fit matter to be relegated to the machines." (Vannevar Bush, "As We May Think", 1945)

"The words of the language, as they are written or spoken, do not seem to play any role in any mechanism of thought. The physical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be 'voluntarily' reproduced or combined. […]  But taken from a psychological viewpoint, this combinatory play seems to be the essential feature in productive thought - before there is any connection with logical construction in words or other kinds of signs which can be communicated to others. The above-mentioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associative play is sufficiently established and can be reproduced at will." (Albert Einstein, [letter to Hadamard, in (Jacques Hadamard, "The Psychology of Invention in the Mathematical Field,1945)])

"For, in mathematics or symbolic logic, reason can crank out the answer from the symboled equations -even a calculating machine can often do so - but it cannot alone set up the equations. Imagination resides in the words which define and connect the symbols - subtract them from the most aridly rigorous mathematical treatise and all meaning vanishes." (Ralph W Gerard, "The Biological Basis of Imagination", American Thought, 1947)

"The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking." (John von Neumann, "The Mathematician" [in "Works of the Mind" Vol. I, 1947])

"Any useful logic must concern itself with Ideas with a fringe of vagueness and a Truth that is a matter of degree." (Norbert Wiener, "Cybernetics", 1948)