Out of Context: On Proofs (Definitions)

"Mathematical proofs are out of the reach of topical arguments; and are not to be attacked by the equivocal use of words or declaration, that make so great a part of other discourses, - nay, even of controversies.” (John Locke, “An Essay Concerning Human Understanding”, 1690)

"[…] it is an error to believe that rigor in the proof is the enemy of simplicity." (David Hilbert, [Paris International Congress] 1900)

"Proof is an idol before whom the pure mathematician tortures himself." (Sir Arthur S Eddington, "The Nature of the Physical World", 1928)

"A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details." (Hermann Weyl, "Unterrichtsblätter für Mathematik und Naturwissenschaften", 1932)

"A mathematical proof is demonstrative reasoning [...]" (George Pólya, "Mathematics and Plausible Reasoning", 1954)

"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing." (George Pólya, "Induction and Analogy in Mathematics", 1954)

"Rigorous proofs are the hallmark of mathematics, they are an essential part of mathematics’ contribution to general culture." (George Pólya, "Mathematical Discovery", 1962)

"[...] the proof is a sequence of actions (applications of rules of inference) that, operating initially on the axioms, transform them into the desired theorem." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

"A proof is a construction that can be looked over, reviewed, verified by a rational agent." (Thomas Tymoczko, "The Four Color Problems", Journal of Philosophy , Vol. 76, 1979)

"Proof serves many purposes simultaneously […] Proof is respectability. Proof is the seal of authority. Proof, in its best instance, increases understanding by revealing the heart of the matter. Proof suggests new mathematics […] Proof is mathematical power, the electric voltage of the subject which vitalizes the static assertions of the theorems." (Reuben Hersh, "The Mathematical Experience", 1981)

"People might suppose that a mathematical proof is conceived as a logical progression, where each step follows upon the ones that have preceded it." (Roger Penrose, "The Emperor’s New Mind", 1989)

"A mathematical proof is a chain of logical deductions, all stemming from a small number of initial assumptions ('axioms') and subject to the strict rules of mathematical logic." (Eli Maor, "e: The Story of a Number", 1994)

"Mathematics rests on proof - and proof is eternal." (Saunders Mac Lane,"Reponses to …", Bulletin of the American Mathematical Society Vol. 30 (2), 1994)

"Proofs are not impersonal, they express the personality of their creator/discoverer just as much as literary efforts do. If something important is true, there will be many reasons that it is true, many proofs of that fact." (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)

"Within mathematics, a proof is an intellectual structure in which premises are conveyed to their conclusions by specific inferential steps. Assumptions in mathematics are called axioms, and conclusions theorems. This definition may be sharpened a little bit. A proof is a finite series of statements such that every statement is either an axiom or follows directly from an axiom by means of tight, narrowly defined rules." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"[…] a proof is a device of communication." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"A proof is a series of steps based on the (adopted) axioms and deduction rules which reaches a desired conclusion." (Cristian S Calude et al, "Proving and Programming", 2007)

"A proof is part of a situational ethic." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"Heuristically, a proof is a rhetorical device for convincing someone else that a mathematical statement is true or valid." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"[…] proof is central to what modern mathematics is about, and what makes it reliable and reproducible." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"So the theorems and propositions are the new heights of knowledge we achieve, while the proofs are essential as they are the mortar which attaches them to the level below. Without proofs the structure would collapse." (Sidney A Morris, "Topology without Tears", 2007)

"A proof is like a piece of theatre or music, with moments of high drama where some major shift takes the audience into a new realm." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"[…] proof is the key ingredient of the emotional side of mathematics; proof is the ultimate explanation of why something is true, and a good proof often has a powerful emotional impact, boosting confidence and encouraging further questions ‘why’." (Alexandre V Borovik, "Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice", 2009)

"A mathematical proof is a watertight argument which begins with information you are given, proceeds by logical argument, and ends with what you are asked to prove." (Sydney A Morris, "Topology without Tears", 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)

"A proof is simply a story. The characters are the elements of the problem, and the plot is up to you." (Paul Lockhart, "Measurement", 2012)

"A mathematical proof is like a battle, or if you prefer a less warlike metaphor, a game of chess. Once a potential weak point has been identified, the mathematician’s technical grasp of the machinery of mathematics can be brought to bear to exploit it." (Ian Stewart, "Visions of Infinity", 2013)

"A proof tells us where to concentrate our doubts. […] An elegantly executed proof is a poem in all but the form in which it is written." (Morris Kline)

"Proofs are to mathematics what spelling (or even calligraphy) is to poetry. Mathematical works do consist of proofs, just as poems do consist of words." (Vladimir Arnold)

On Reasoning: Demonstrative vs Plausible Reasoning

"In metaphysical reasoning, the process is always short. The conclusion is but a step or two, seldom more, from the first principles or axioms on which it is grounded, and the different conclusions depend not one upon another.
It is otherwise in mathematical reasoning. Here the field has no limits. One proposition leads on to another, that to a third, and so on without end. If it should be asked, why demonstrative reasoning has so wide a field in mathematics, while, in other abstract subjects, it is confined within very narrow limits, I conceive this is chiefly owing to the nature of quantity, […] mathematical quantities being made up of parts without number, can touch in innumerable points, and be compared in innumerable different ways." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"Mathematicians have, in many cases, proved some things to be possible and others to be impossible, which, without demonstration, would not have been believed […] Mathematics afford many instances of impossibilities in the nature of things, which no man would have believed, if they had not been strictly demonstrated. Perhaps, if we were able to reason demonstratively in other subjects, to as great extent as in mathematics, we might find many things to be impossible, which we conclude, without hesitation, to be possible." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"The mathematician pays not the least regard either to testimony or conjecture, but deduces everything by demonstrative reasoning, from his definitions and axioms. Indeed, whatever is built upon conjecture, is improperly called science; for conjecture may beget opinion, but cannot produce knowledge." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

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

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

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

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

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

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

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

On Algorithms I

"Mathematics is an aspect of culture as well as a collection of algorithms." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"An algorithm must be seen to be believed, and the best way to learn what an algorithm is all about is to try it." (Donald E Knuth, The Art of Computer Programming Vol. I, 1968)

"Scientific laws give algorithms, or procedures, for determining how systems behave. The computer program is a medium in which the algorithms can be expressed and applied. Physical objects and mathematical structures can be represented as numbers and symbols in a computer, and a program can be written to manipulate them according to the algorithms. When the computer program is executed, it causes the numbers and symbols to be modified in the way specified by the scientific laws. It thereby allows the consequences of the laws to be deduced." (Stephen Wolfram, "Computer Software in Science and Mathematics", 1984)

"Algorithmic complexity theory and nonlinear dynamics together establish the fact that determinism reigns only over a quite finite domain; outside this small haven of order lies a largely uncharted, vast wasteland of chaos." (Joseph Ford, "Progress in Chaotic Dynamics: Essays in Honor of Joseph Ford's 60th Birthday", 1988)

"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)

"Algorithms are a set of procedures to generate the answer to a problem." (Stuart Kauffman, "At Home in the Universe: The Search for Laws of Complexity", 1995)

"Let us regard a proof of an assertion as a purely mechanical procedure using precise rules of inference starting with a few unassailable axioms. This means that an algorithm can be devised for testing the validity of an alleged proof simply by checking the successive steps of the argument; the rules of inference constitute an algorithm for generating all the statements that can be deduced in a finite number of steps from the axioms." (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)

"Heuristics are rules of thumb that help constrain the problem in certain ways (in other words they help you to avoid falling back on blind trial and error), but they don't guarantee that you will find a solution. Heuristics are often contrasted with algorithms that will guarantee that you find a solution - it may take forever, but if the problem is algorithmic you will get there. However, heuristics are also algorithms." (S Ian Robertson, "Problem Solving", 2001)

"An algorithm is a simple rule, or elementary task, that is repeated over and over again. In this way algorithms can produce structures of astounding complexity." (F David Peat, "From Certainty to Uncertainty", 2002)

"Many people have strong intuitions about whether they would rather have a vital decision about them made by algorithms or humans. Some people are touchingly impressed by the capabilities of the algorithms; others have far too much faith in human judgment. The truth is that sometimes the algorithms will do better than the humans, and sometimes they won’t. If we want to avoid the problems and unlock the promise of big data, we’re going to need to assess the performance of the algorithms on a case-by-case basis. All too often, this is much harder than it should be. […] So the problem is not the algorithms, or the big datasets. The problem is a lack of scrutiny, transparency, and debate." (Tim Harford, "The Data Detective: Ten easy rules to make sense of statistics", 2020)

On Structure: Structure in Mathematics III

"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)

"The probability concept used in probability theory has exactly the same structure as have the fundamental concepts in any field in which mathematical analysis is applied to describe and represent reality." (Richard von Mises, "Mathematical Theory of Probability and Statistics", 1964)

"The question ‘What is mathematics?’ cannot be answered meaningfully by philosophical generalities, semantic definitions or journalistic circumlocutions. Such characterizations also fail to do justice to music or painting. No one can form an appreciation of these arts without some experience with rhythm, harmony and structure, or with form, color and composition. For the appreciation of mathematics actual contact with its substance is even more necessary." (Richard Courant, "Mathematics in the Modern World", Scientific American Vol. 211 (3), 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)

"The most natural way to give an independence proof is to establish a model with the required properties. This is not the only way to proceed since one can attempt to deal directly and analyze the structure of proofs. However, such an approach to set theoretic questions is unnatural since all our intuition come from our belief in the natural, almost physical model of the mathematical universe." (Paul J Cohen, "Set Theory and the Continuum Hypothesis", 1966)

"The structures with which mathematics deals are more like lace, the leaves of trees, and the play of light and shadow on a human face, than they are like buildings and machines, the least of their representatives. The best proofs in mathematics are short and crisp like epigrams, and the longest have swings and rhythms that are like music. The structures of mathematics and the propositions about them are ways for the imagination to travel and the wings, or legs, or vehicles to take you where you want to go." (Scott Buchanan, "Poetry and Mathematics", 1975)

"In each branch of mathematics it is essential to recognize when two structures are equivalent. For example two sets are equivalent, as far as set theory is concerned, if there exists a bijective function which maps one set onto the other. Two groups are equivalent, known as isomorphic, if there exists a a homomorphism of one to the other which is one-to-one and onto. Two topological spaces are equivalent, known as homeomorphic, if there exists a homeomorphism of one onto the other." (Sydney A Morris, "Topology without Tears", 2011)

On Imagination (2000-2024)

"One of the most fundamental notions in mathematics is that of number. Although the idea of number is basic, the numbers themselves possess both nuance and complexity that spark the imagination." (Edward B Burger, "Exploring the Number Jungle", 2000)

"To say that a thing is imaginary is not to dispose of it in the realm of mind, for the imagination, or the image making faculty, is a very important part of our mental functioning. An image formed by the imagination is a reality from the point of view of psychology; it is quite true that it has no physical existence, but are we going to limit reality to that which is material? We shall be far out of our reckoning if we do, for mental images are potent things, and although they do not actually exist on the physical plane, they influence it far more than most people suspect." (Dion Fortune," Spiritualism and Occultism", 2000)

"Science begins with the world we have to live in, accepting its data and trying to explain its laws. From there, it moves toward the imagination: it becomes a mental construct, a model of a possible way of interpreting experience. The further it goes in this direction, the more it tends to speak the language of mathematics, which is really one of the languages of the imagination, along with literature and music." (Northrop Frye, "The Educated Imagination", 2002)

"[…] because observations are all we have, we take them seriously. We choose hard data and the framework of mathematics as our guides, not unrestrained imagination or unrelenting skepticism, and seek the simplest yet most wide-reaching theories capable of explaining and predicting the outcome of today’s and future experiments." (Brian Greene, "The Fabric of the Cosmos", 2004)

"There is a strong parallel between mountain climbing and mathematics research. When first attempts on a summit are made, the struggle is to find any route. Once on the top, other possible routes up may be discerned and sometimes a safer or shorter route can be chosen for the descent or for subsequent ascents. In mathematics the challenge is finding a proof in the first place. Once found, almost any competent mathematician can usually find an alternative often much better and shorter proof. At least in mountaineering we know that the mountain is there and that, if we can find a way up and reach the summit, we shall triumph. In mathematics we do not always know that there is a result, or if the proposition is only a figment of the imagination, let alone whether a proof can be found." (Kathleen Ollerenshaw, "To talk of many things: An autobiography", 2004)

"Imagination has the creative task of making symbols, joining things together in such a way that they throw new light on each other and on everything around them. The imagination is a discovering faculty, a faculty for seeing relationships, for seeing meanings that are special and even quite new." (Thomas Merton, "Angelic Mistakes: The Art of Thomas Merton", 2006)

"To have the courage to think outside the square, we need to be intrigued by a problem. This intrigue will encourage us to use our imaginations to find solutions which are beyond our current view of the world. This was the challenge that faced mathematicians as they searched for a solution to the problem of finding meaning for the square root of a negative number, in particular v-1." (Les Evans, "Complex Numbers and Vectors", 2006)

"Unfortunately, if we were to use geometry to explore the concept of the square root of a negative number, we would be setting a boundary to our imagination that would be difficult to cross. To represent -1 using geometry would require us to draw a square with each side length being less than zero. To be asked to draw a square with side length less than zero sounds similar to the Zen Buddhists asking ‘What is the sound of one hand clapping?’" (Les Evans, "Complex Numbers and Vectors", 2006)

"Language use is a curious behavior, but once the transition to language is made, literature is a likely consequence, since it is linked to the dynamic of the linguistic symbol through the functioning of the imagination." (Russell Berman, "Fiction Sets You Free: Literature, Liberty and Western Culture", 2007)

"If worldviews or metanarratives can be compared to lenses, which of them brings things into the sharpest focus? This is not an irrational retreat from reason. Rather, it is about grasping a deeper order of things which is more easily accessed by the imagination than by reason." (Alister McGrath, "If I Had Lunch with C. S. Lewis: Exploring the Ideas of C. S. Lewis on the Meaning of Life", 2014)

"Mathematics is a fascinating discipline that calls for creativity, imagination, and the mastery of rigorous standards of proof." (John Meier & Derek Smith, "Exploring Mathematics: An Engaging Introduction to Proof", 2017)

"The mental model is the arena where imagination takes place. It enables us to experiment with different scenarios by making local alterations to the model. […] To speak of causality, we must have a mental model of the real world. […] Our shared mental models bind us together into communities." (Judea Pearl & Dana Mackenzie, "The Book of Why: The new science of cause and effect", 2018)

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On Conjecture (1975-1999)

"All knowledge, the sociologist could say, is conjectural and theoretical. Nothing is absolute and final. Therefore all knowledge is relative to the local situation of the thinkers who produce it: the ideas and conjectures that they are capable of producing: the problems that bother them; the interplay of assumptions and criticism in their milieu; their purposes and aims; the experiences they have and the standards and meanings they apply." (David Bloor, "Knowledge and Social Imagery", 1976)

"The essential function of a hypothesis consists in the guidance it affords to new observations and experiments, by which our conjecture is either confirmed or refuted." (Ernst Mach, "Knowledge and Error: Sketches on the Psychology of Enquiry", 1976)

"The verb 'to theorize' is now conjugated as follows: 'I built a model; you formulated a hypothesis; he made a conjecture.'" (John M Ziman, "Reliable Knowledge", 1978)

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

"So-called scientific knowledge is not knowledge, for it consists only of conjectures or hypotheses - even if some have gone through the crossfire of ingenious tests." (Karl R Popper, "Epistemology and the Problem of Peace", [lecture in "All Life is Problem Solving", 1999] 1985)

"Three shifts can be detected over time in the understanding of mathematics itself. One is a shift from completeness to incompleteness, another from certainty to conjecture, and a third from absolutism to relativity." (Leone Burton, "Femmes et Mathematiques: Y a–t–il une?",  Association for Women in Mathematics Newsletter, Intersection 18, 1988)

"A mathematical proof is a chain of logical deductions, all stemming from a small number of initial assumptions ('axioms') and subject to the strict rules of mathematical logic. Only such a chain of deductions can establish the validity of a mathematical law, a theorem. And unless this process has been satisfactorily carried out, no relation - regardless of how often it may have been confirmed by observation - is allowed to become a law. It may be given the status of a hypothesis or a conjecture, and all kinds of tentative results may be drawn from it, but no mathematician would ever base definitive conclusions on it. (Eli Maor, "e: The Story of a Number", 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 …", Bulletin of the American Mathematical Society Vol. 30 (2), 1994)

"The methods of science include controlled experiments, classification, pattern recognition, analysis, and deduction. In the humanities we apply analogy, metaphor, criticism, and (e)valuation. In design we devise alternatives, form patterns, synthesize, use conjecture, and model solutions." (Béla H Bánáthy, "Designing Social Systems in a Changing World", 1996)

"A proof of a mathematical theorem is a sequence of steps which leads to the desired conclusion. The rules to be followed [...] were made explicit when logic was formalized early in the this century [...] These rules can be used to disprove a putative proof by spotting logical errors; they cannot, however, be used to find the missing proof of a [...] conjecture. [...] Heuristic arguments are a common occurrence in the practice of mathematics. However... The role of heuristic arguments has not been acknowledged in the philosophy of mathematics despite the crucial role they play in mathematical discovery. [...] Our purpose is to bring out some of the features of mathematical thinking which are concealed beneath the apparent mechanics of proof." (Gian-Carlo Rota, "Indiscrete Thoughts", 1997)

"Architectural conjectures are mathematically precise assertions, as well milled as minted coins, provisionally usable in the commerce of logical arguments; less than ‘coins’ and more aptly, promissory notes to be paid in full by some future demonstration, or to be contradicted. These conjectures are expected to turn out to be true, as, of course, are all conjectures; their formulation is often away of "formally" packaging, or at least acknowledging, an otherwise shapeless body of mathematical experience that points to their truth." (Barry Mazur, "Conjecture", Synthese 111, 1997)

"The everyday usage of 'theory' is for an idea whose outcome is as yet undetermined, a conjecture, or for an idea contrary to evidence. But scientists use the word in exactly the opposite sense. [In science] 'theory' [...] refers only to a collection of hypotheses and predictions that is amenable to experimental test, preferably one that has been successfully tested. It has everything to do with the facts." (Tony Rothman & George Sudarshan, "Doubt and Certainty: The Celebrated Academy: Debates on Science, Mysticism, Reality, in General on the Knowable and Unknowable", 1998)

"A mathematician experiments, amasses information, makes a conjecture, finds out that it does not work, gets confused and then tries to recover. A good mathematician eventually does so - and proves a theorem." (Steven Krantz, "Conformal Mappings", American Scientist, 1999)

Thomas Stark - Collected Quotes

"Basis real and imaginary numbers have eternal and necessary reality. Complex numbers do not. They are temporal and contingent in the sense that for complex numbers to exist, we first have to carry out an operation: adding basis real and imaginary numbers together. Complex numbers therefore do not exist in their own right. They are constructed. They are derived. Symmetry breaking is exactly where constructed numbers come into existence. The very act of adding a sine wave to a cosine wave is the sufficient condition to create a broken symmetry: a complex number. The 'Big Bang', mathematically, is simply where a perfect array of basis sine and cosine waves start entering into linear combinations, creating a chain reaction, an 'explosion', of complex numbers - which corresponds to the “physical” universe." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

"Euler’s formula - although deceptively simple - is actually staggeringly conceptually difficult to apprehend in its full glory, which is why so many mathematicians and scientists have failed to see its extraordinary scope, range, and ontology, so powerful and extensive as to render it the master equation of existence, from which the whole of mathematics and science can be derived, including general relativity, quantum mechanics, thermodynamics, electromagnetism and the strong and weak nuclear forces! It’s not called the God Equation for nothing. It is much more mysterious than any theistic God ever proposed." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

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

"It is in fact mathematics itself that is simplest in hypothesis and also richest in phenomena (i.e. the simple source of all complexity). In ontological mathematics, all of existence comprises sinusoidal waves arranged into autonomous units called monads, and these are all that are required to explain everything." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

"Mathematics is existence itself, existence in itself, existence for itself. […] Mathematics is the basis of everything. Mathematics is the true God, and it creates the universe from itself, using, naturally, mathematics." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

"Ontological mathematics is operating in such a way as to organize itself into a zero-entropy structure – mathematical perfection. The 'Big Bang' is equivalent to the total scrambling of a cosmic Rubik’s Cube. The task of ontological mathematics is then to unscramble the Cube and return it to its original, pristine configuration. Emotionally, this amounts to returning to perfect Love and Bliss. Intellectually, it means reaching a state of perfect logic and reason [...] thinking perfectly." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

"Reason is indeed all about identity, or, rather, tautology. Mathematics is the eternal, necessary system of rational, analytic tautology. Tautology is not 'empty', as it is so often characterized by philosophers. It is in fact the fullest thing there, the analytic ground of existence, and the basis of everything. Mathematical tautology has infinite masks to wear, hence delivers infinite variety. Mathematical tautology provides Leibniz’s world that is 'simplest in hypothesis and the richest in phenomena'. No hypothesis cold be simpler than the one revolving around tautologies concerning 'nothing'. There is something - existence - because nothing is tautologous, and 'something' is how that tautology is expressed. If we write x = 0, where x is any expression that has zero as its net result, then we have a world of infinite possibilities where something ('x') equals nothing (0)." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

"The laws of the universe cannot be different from the universe. The laws of the universe must actually be the universe. Otherwise you create an impossible Cartesian dualism of laws and things operated on by laws. How can a law operate on a non-law? That’s a category error. The only way to make the laws and 'stuff' of the universe the same is via mathematics." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

"The scientific method does not begin with the injunction to use reason and logic, and to obey the principle of sufficient reason and Occam’s razor. Instead, it begins with the word 'Observe'. In other words, if reality in itself is unobservable - which is of course the case - then the scientific method automatically fails to tell us a single thing about it." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

"There are two things for which you would never use science if you wished to prove their existence. One is God and the other is mathematics. No one would ever turn to scientific arguments to establish the existence of God, and, equally, no one would ever appeal to science to explain the ontology of mathematics." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

"There is no such thing as randomness. No one who could detect every force operating on a pair of dice would ever play dice games, because there would never be any doubt about the outcome. The randomness, such as it is, applies to our ignorance of the possible outcomes. It doesn’t apply to the outcomes themselves. They are 100% determined and are not random in the slightest. Scientists have become so confused by this that they now imagine that things really do happen randomly, i.e. for no reason at all." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

"Zero is not a point of non-existence. Zero is always a balance point of existents. The human understanding of 'zero' must undergo the most radical of all transformations. Most people, especially scientists, associate it with absolute nothingness, with non-existence. This is absolutely untrue. Or, to put it another way, we can define it in two ways: 1) nothing as non-existence, in which case it has absolutely no consequences but leads to all manner of abstract paradoxes and contradictions, or 2) nothing as existence, in which case it is always a mathematical balance point for somethings. It is purely mathematical, not scientific, or religious, or spiritual, or emotional, or sensory, or mystical. It is analytic nothing and whenever you encounter it you have to establish the exact means by which it is maintaining its balance of zero." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

Set Theory III

"One very important genus of complex ideas that we encounter everywhere are those in which the idea of collection (Inbegriff ) appears. There are many types of the latter [...] I must first determine with more precision the concept I associate with the word collection. I use this word in the same sense as it is used in the common usage and thus understand by a collection of certain things exactly the same as what one would express by the words: a combination (Verbindung) or association (Vereinigung) of these things, a gathering (Zusammensein) of the latter, a whole (Ganzes) in which they occur as parts (Teile). Hence the mere idea of a collection does not allow us to determine in which order and sequence the things that are put together appear or, indeed, whether there is or can be such an order. [...] A collection, it seems to me, is nothing other than something complex (das Zusammengesetztheit hat)." (Bernard Bolzano, "Wissenschaftslehre" ["Theory of Science"], 1837)

"The old and oft-repeated proposition 'Totum est majus sua parte' [the whole is larger than the part] may be applied without proof only in the case of entities that are based upon whole and part; then and only then is it an undeniable consequence of the concepts 'totum' and 'pars'. Unfortunately, however, this 'axiom' is used innumerably often without any basis and in neglect of the necessary distinction between 'reality' and 'quantity' , on the one hand, and 'number' and 'set', on the other, precisely in the sense in which it is generally false." (Georg Cantor, "Über unendliche, lineare Punktmannigfaltigkeiten", Mathematische Annalen 20, 1882)

"The foregoing account of my researches in the theory of manifolds has reached a point where further progress depends on extending the concept of true integral number beyond the previous boundaries; this extension lies in a direction which, to my knowledge, no one has yet attempted to explore.
My dependence on this extension of number concept is so great, that without it I should be unable to take freely the smallest step further in the theory of sets." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883) 

"To the thought of considering the infinitely great not merely in the form of what grows without limits - and in the closely related form of the convergent infinite series first introduced in the seventeenth century-, but also fixing it mathematically by numbers in the determinate form of the completed-infinite, I have been logically compelled in the course of scientific exertions and attempts which have lasted many years, almost against my will, for it contradicts traditions which had become precious to me; and therefore I believe that no arguments can be made good against it which I would not know how to meet." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883)

"Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. [...] The student's task in learning set theory is to steep himself in unfamiliar but essentially shallow generalities till they become so familiar that they can be used with almost no conscious effort. In other words, general set theory is pretty trivial stuff really, but, if you want to be a mathematician, you need some, and here it is; read it, absorb it, and forget it [...] the language and notation are those of ordinary informal mathematics. A more important way in which the naive point of view predominates is that set theory is regarded as a body of facts, of which the axioms are a brief and convenient summary; in the orthodox axiomatic view the logical relations among various axioms are the central objects of study." (Paul R Halmos, "Naive Set Theory", 1960)

"A manifold, roughly, is a topological space in which some neighborhood of each point admits a coordinate system, consisting of real coordinate functions on the points of the neighborhood, which determine the position of points and the topology of that neighborhood; that is, the space is locally cartesian. Moreover, the passage from one coordinate system to another is smooth in the overlapping region, so that the meaning of 'differentiable' curve, function, or map is consistent when referred to either system." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"Set theory is concerned with abstract objects and their relation to various collections which contain them. We do not define what a set is but accept it as a primitive notion. We gain an intuitive feeling for the meaning of sets and, consequently, an idea of their usage from merely listing some of the synonyms: class, collection, conglomeration, bunch, aggregate. Similarly, the notion of an object is primitive, with synonyms element and point. Finally, the relation between elements and sets, the idea of an element being in a set, is primitive." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"The mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"Does set theory, once we get beyond the integers, refer to an existing reality, or must it be regarded, as formalists would regard it, as an interesting formal game? [...] A typical argument for the objective reality of set theory is that it is obtained by extrapolation from our intuitions of finite objects, and people see no reason why this has less validity. Moreover, set theory has been studied for a long time with no hint of a contradiction. It is suggested that this cannot be an accident, and thus set theory reflects an existing reality. In particular, the Continuum Hypothesis and related statements are true or false, and our task is to resolve them." (Paul Cohen, "Skolem and pessimism about proof in mathematics", Philosophical Transactions of the Royal Society A 363 (1835), 2005)

"In each branch of mathematics it is essential to recognize when two structures are equivalent. For example two sets are equivalent, as far as set theory is concerned, if there exists a bijective function which maps one set onto the other. Two groups are equivalent, known as isomorphic, if there exists a a homomorphism of one to the other which is one-to-one and onto. Two topological spaces are equivalent, known as homeomorphic, if there exists a homeomorphism of one onto the other." (Sydney A Morris, "Topology without Tears", 2011)

Imre Lakatos - Collected Quotes

"No experimental result can ever kill a theory: any theory can be saved from counterinstances either by some auxiliary hypothesis or by a suitable reinterpretation of its terms." (Imre Lakatos, "Falsification and the Methodology of Scientific Research Programmes", 1965)

"Blind commitment to a theory is not an intellectual virtue: it is an intellectual crime." (Imre Lakatos, [radio Lecture] 1973)

"Mathematics does not grow through a monotonous increase of the number of indubitably established theorems, but through the incessant improvement of guesses by speculation and criticism." (Imre Lakatos, "Proofs and Refutations", 1976)

"There is an infinite regress in proofs; therefore proofs do not prove. You should realize that proving is a game, to be played while you enjoy it and stopped when you get tired of it." (Imre Lakatos, "Proofs and Refutations", 1976)

"Under the present dominance of formalism, one is tempted to paraphrase Kant: the history of mathematics, lacking the guidance of philosophy, has become blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the of mathematics, has become empty." (Imre Lakatos, "Proofs and Refutations: The Logic of Mathematical Discovery", 1976)

"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)

"Mathematics has been trivialized, derived from indubitable, trivial axioms in which only absolutely clear trivial terms figure, and from which truth pours down in clear channels." (Imre Lakatos, "Mathematics, Science and Epistemology", 1980)

"Philosophy of science without history of science is empty; history of science without philosophy of science is blind." (Imre Lakatos, "The Methodology of Scientific Research Programmes" Vol. 1, 1980)

"There is no falsification before the emergence of a better theory." (Imre Lakatos, "The Methodology of Scientific Research Programmes" Vol. 1, 1980)

On Axioms (1600-1699)

"It cannot be that axioms established by argumentation should avail for the discovery of new works, since the subtlety of nature is greater many times over than the subtlety of argument. But axioms duly and orderly formed from particulars easily discover the way to new particulars, and thus render sciences active." (Francis Bacon, "Novum Organum", 1620)

"There are and can be only two ways of searching into and discovering truth. The one flies from the senses and particulars to the most general axioms, and from these principles, the truth of which it takes for settled and immovable, proceeds to judgment and to the discovery of middle axioms. And this way is now in fashion. The other derives axioms from the senses and particulars, rising by a gradual and unbroken ascent, so that it arrives at the most general axioms last of all. This is the true way, but as yet untried." (Francis Bacon, "Novum Organum", 1620)

"We must first, by every kind of experiment, elicit the discovery of causes and true axioms, and seek for experiments which may afford light rather than profit." (Francis Bacon, "Novum Organum", 1620)

"Rules for Axioms. I. Not to omit any necessary principle without asking whether it is admittied, however clear and evident it may be. II. Not to demand, in axioms, any but things that are perfectly evident in themselves." (Blaise Pascal, "The Art of Persuasion", cca. 1658)

"For it is unquestionable that it is no great error to define and clearly explain things, although very clear of themselves, nor to omit to require in advance axioms which cannot be refused in the place where they are necessary; nor lastly to prove propositions that would be admitted without proof." (Blaise Pascal, "The Art of Persuasion",  cca. 1658)

"To prove all propositions, and to employ nothing for their proof but axioms fully evident of themselves, or propositions already demonstrated or admitted; Never to take advantage of the ambiguity of terms by failing mentally to substitute definitions that restrict or explain them." (Blaise Pascal, "The Art of Persuasion", cca. 1658)

"This art, which I call the art of persuading, and which, properly speaking, is simply the process of perfect methodical proofs, consists of three essential parts: of defining the terms of which we should avail ourselves by clear definitions, of proposing principles of evident axioms to prove the thing in question; and of always mentally substituting in the demonstrations the definition in the place of the thing defined." (Blaise Pascal, "The Art of Persuasion", cca. 1658)

"Mathematicians who are only mathematicians have exact minds, provided all things are explained to them by means of definitions and axioms; otherwise they are inaccurate and insufferable, for they are only right when the principles are quite clear." (Blaise Pascal, "Pensées", 1670)

"Rules necessary for definitions. Not to leave any terms at all obscure or ambiguous without definition; Not to employ in definitions any but terms perfectly known or already explained. […] A few rules include all that is necessary for the perfection of the definitions, the axioms, and the demonstrations, and consequently of the entire method of the geometrical proofs of the art of persuading." (Blaise Pascal, "Pensées", 1670)

On Axioms (1900-1909)

"If geometry is to serve as a model for the treatment of physical axioms, we shall try first by a small number of axioms to include as large a class as possible of physical phenomena, and then by adjoining new axioms to arrive gradually at the more special theories. […] The mathematician will have also to take account not only of those theories coming near to reality, but also, as in geometry, of all logically possible theories. We must be always alert to obtain a complete survey of all conclusions derivable from the system of axioms assumed." (David Hilbert, 1900)

"When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas; and no statement within the realm of the science... is held to be correct unless it can be derived from axioms by means of a finite number of logical steps. Upon closer consideration the question arises: Whether, in any way, certain statements of single axioms depend upon one another, and whether the axioms may not therefore contain certain parts in common, which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another." (David Hilbert, "Mathematische Probleme", Gŏttinger Nachrichten, 1900)

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

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

"Like almost every subject of human interest, this one [mathematics] is just as easy or as difficult as we choose to make it. A lifetime may be spent by a philosopher in discussing the truth of the simplest axiom. The simplest fact as to our existence may fill us with such wonder that our minds will remain overwhelmed with wonder all the time." (John Perry, "Teaching of Mathematics", 1902)

"No theorem can be new unless a new axiom intervenes in its demonstration; reasoning can only give us immediately evident truths borrowed from direct intuition; it would only be an intermediary parasite." (Henri Poincaré, "Science and Hypothesis", 1902)

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

"Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted." (Henri Poincaré," Science and Method", 1908)

"It has been argued that mathematics is not or, at least, not exclusively an end in itself; after all it should also be applied to reality. But how can this be done if mathematics consisted of definitions and analytic theorems deduced from them and we did not know whether these are valid in reality or not. One can argue here that of course one first has to convince oneself whether the axioms of a theory are valid in the area of reality to which the theory should be applied. In any case, such a statement requires a procedure which is outside logic." (Ernst Zermelo, "Mathematische Logik - Vorlesungen gehalten von Prof. Dr. E. Zermelo zu Göttingen im S. S", 1908)

"It is by logic that we prove, but by intuition that we discover. [...] Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted." (Henri Poincaré, "Science and Method", 1908)

"I do in no wise share this view [that the axioms are arbitrary propositions which we assume wholly at will, and that in like manner the fundamental conceptions are in the end only arbitrary symbols with which we operate] but consider it the death of all science: in my judgment the axioms of geometry are not arbitrary, but reasonable propositions which generally have the origin in space intuition and whose separate content and sequence is controlled by reasons of expediency." (Felix Klein, "Elementarmathematik vom hoheren Standpunkte aus", 1909)

On Imagination (1800-1849)

"The philosopher who is really useful to the cause of science, is he who, uniting to a fertile imagination, a rigid severity in investigation and observation, is at once tormented by the desire of ascertaining the cause of the phenomena, and by the fear of deceiving himself in that which he assigns." (Pierre-Simon Laplace, "System of the World" Vol. 2, 1809)

"When the eye or the imagination is struck with an uncommon work, the next transition of an active mind is to the means by which it was performed." (Samuel Johnson, 1810)

"The imagination […] that reconciling and mediatory power, which incorporating the reason in images of the sense and organizing (as it were) the flux of the senses by the permanence and self-circling energies of the reason, gives birth to a system of symbols, harmonious in themselves, and consubstantial with the truths of which they are the conductors." (Samuel T Coleridge, "The Statesman's Manual", 1816)

"It seems to be like taking the pieces of a dissected map out of its box. We first look at one part, and then at another, then join and dove-tail them; and when the successive acts of attention have been completed, there is a retrogressive effort of mind to behold it as a whole. The poet should paint to the imagination, not to the fancy; and I know no happier case to exemplify the distinction between these two faculties." (Samuel T Coleridge," Biographia Literaria", 1817)

"Whilst chemical pursuits exalt the understanding, they do not depress the imagination or weaken genuine feeling; whilst they give the mind habits of accuracy, by obliging it to attend to facts, they likewise extend its analogies; and, though conversant with the minute forms of things, they have for their ultimate end the great and magnificent objects of nature." (Sir Humphry Davy, "Consolations in Travel, or the Last Days of a Philosopher", 1830)

"No occupation is more worthy of an intelligent and enlightened mind, than the study of Nature and natural objects; and whether we labour to investigate the structure and function of the human system, whether we direct our attention to the classification and habits of the animal kingdom, or prosecute our researches in the more pleasing and varied field of vegetable life, we shall constantly find some new object to attract our attention, some fresh beauties to excite our imagination, and some previously undiscovered source of gratification and delight." (Sir Joseph Paxton, "A Practical Treatise on the Cultivation of the Dahlia", 1838)

"But a thousand unconnected observations have no more value, as a demonstrative proof, than a single one. If we do not succeed in discovering causes by our researches, we have no right to create them by the imagination; we must not allow mere fancy to proceed beyond the bounds of our knowledge."(Justus von Liebig, "The Lancet", 1844)

"The nose of a mob is its imagination. By this, at any time, it can be quietly led." (Edgar A Poe, "The Works of Edgar Allan Poe", 1849)

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On Imagination (1900-1924)

"This is the greatest degree of impoverishment; the [mental] image, deprived little by little of its own characteristics, is nothing more than a shadow. […] Being dependent on the state of the brain, the image undergoes change like all living substance, - it is subject to gains and losses, especially losses. But each of the foregoing three classes has its use for the inventor. They serve as material for different kinds of imagination - in their concrete form, for the mechanic and the artist; in their schematic form, for the scientist and for others." (Théodule-Armand Ribot, "Essay on the Creative Imagination", 1900)

"This means that it is not a dead thing; it is not at all like a photographic plate with which one may reproduce copies indefinitely. Being dependent on the state of the brain, the image undergoes change like all living substance, - it is subject to gains and losses, especially losses. But each of the foregoing three classes has its use for the inventor. They serve as material for different kinds of imagination - in their concrete form, for the mechanic and the artist; in their schematic form, for the scientist and for others." (Théodule-Armand Ribot, "Essay on the Creative Imagination" , 1900)

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

"Imagination is as vital to any advance in science as learning and precision are essential for starting points." (Percival Lowell, "The Solar System", 1903)

"Nature talks in symbols; he who lacks imagination cannot understand her." (Abraham Miller, "Unmoral Maxims", 1906)

"Mathematics makes constant demands upon the imagination, calls for picturing in space (of one, two, three dimensions), and no considerable success can be attained without a growing ability to imagine all the various possibilities of a given case, and to make them defile before the mind's eye." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"The motive for the study of mathematics is insight into the nature of the universe. Stars and strata, heat and electricity, the laws and processes of becoming and being, incorporate mathematical truths. If language imitates the voice of the Creator, revealing His heart, mathematics discloses His intellect, repeating the story of how things came into being. And the value of mathematics, appealing as it does to our energy and to our honor, to our desire to know the truth and thereby to live as of right in the household of God, is that it establishes us in larger and larger certainties. As literature develops emotion, understanding, and sympathy, so mathematics develops observation, imagination, and reason." (William E Chancellor, "A Theory of Motives, Ideals and Values in Education" 1907)

"The beautiful has its place in mathematics as elsewhere. The prose of ordinary intercourse and of business correspondence might be held to be the most practical use to which language is put, but we should be poor indeed without the literature of imagination. Mathematics too has its triumphs of the Creative imagination, its beautiful theorems, its proofs and processes whose perfection of form has made them classic. He must be a 'practical' man who can see no poetry in mathematics." (Wiliam F White, "A Scrap-book of Elementary Mathematics: Notes, Recreations, Essays", 1908)

"No system would have ever been framed if people had been simply interested in knowing what is true, whatever it may be. What produces systems is the interest in maintaining against all comers that some favourite or inherited idea of ours is sufficient and right. A system may contain an account of many things which, in detail, are true enough; but as a system, covering infinite possibilities that neither our experience nor our logic can prejudge, it must be a work of imagination and a piece of human soliloquy: It may be expressive of human experience, it may be poetical; but how should anyone who really coveted truth suppose that it was true?" (George Santayana, "The Genteel Tradition in American Philosophy", 1911)

"Only in men’s imagination does every truth find an effective and undeniable existence." (Joseph Conrad, "Some Reminiscences", 1912)

"What is the imagination? Only an arm or weapon of the interior energy; only the precursor of the reason." (Ralph W Emerson, "Miscellanies, Natural history of intellect", 1912)

"The concept of an independent system is a pure creation of the imagination. For no material system is or can ever be perfectly isolated from the rest of the world. Nevertheless it completes the mathematician’s ‘blank form of a universe’ without which his investigations are impossible. It enables him to introduce into his geometrical space, not only masses and configurations, but also physical structure and chemical composition." (Lawrence J Henderson, "The Order of Nature: An Essay", 1917)

"[…] because mathematics contains truth, it extends its validity to the whole domain of art and the creatures of the constructive imagination." (James B Shaw, "Lectures on the Philosophy of Mathematics", 1918)

"Nature uses human imagination to lift her work of creation to even higher levels." (Luigi Pirandello, "Six Characters in Search of an Author", 1921)

"The story of scientific discovery has its own epic unity - a unity of purpose and endeavour - the single torch passing from hand to hand through the centuries; and the great moments of science when, after long labour, the pioneers saw their accumulated facts falling into a significant order - sometimes in the form of a law that revolutionised the whole world of thought - have an intense human interest, and belong essentially to the creative imagination of poetry." (Alfred Noyes, "Watchers of the Sky", 1922)

On Imagination (1950-1974)

"[…] observation is not enough, and it seems to me that in science, as in the arts, there is very little worth having that does not require the exercise of intuition as well as of intelligence, the use of imagination as well as of information." (Kathleen Lonsdale, "Facts About Crystals", American Scientist Vol. 39 (4), 1951)

"There is always an analogy between nature and the imagination, and possibly poetry is merely the strange rhetoric of that parallel." (Wallace Stevens, "The Necessary Angel", 1951)

"All great discoveries in experimental physics have been due to the intuition of men who made free use of models, which were for them not products of the imagination, but representatives of real things." (Max Born, "Physical Reality", Philosophical Quarterly Vol. (11), 1953)

"The creative act owes little to logic or reason. In their accounts of the circumstances under which big ideas occurred to them, mathematicians have often mentioned that the inspiration had no relation to the work they happened to be doing. Sometimes it came while they were traveling, shaving or thinking about other matters. The creative process cannot be summoned at will or even cajoled by sacrificial offering. Indeed, it seems to occur most readily when the mind is relaxed and the imagination roaming freely." (Morris Kline, Scientific American, 1955)

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

"The ultimate origin of the difficulty lies in the fact (or philosophical principle) that we are compelled to use the words of common language when we wish to describe a phenomenon, not by logical or mathematical analysis, but by a picture appealing to the imagination. Common language has grown by everyday experience and can never surpass these limits. Classical physics has restricted itself to the use of concepts of this kind; by analysing visible motions it has developed two ways of representing them by elementary processes; moving particles and waves. There is no other way of giving a pictorial description of motions - we have to apply it even in the region of atomic processes, where classical physics breaks down." (Max Born, "Atomic Physics", 1957)

"In imagination there exists the perfect mystery story. Such a story presents all the essential clews, and compels us to form our own theory of the case. If we follow the plot carefully we arrive at the complete solution for ourselves just before the author’s disclosure at the end of the book. The solution itself, contrary to those of inferior mysteries, does not disappoint us; moreover, it appears at the very moment we expect it." (Leopold Infeld, "The Evolution of Physics", 1961)

"The structures of mathematics and the propositions about them are ways for the imagination to travel and the wings, or legs, or vehicles to take you where you want to go." (Scott Buchanan, "Poetry and Mathematics", 1962)

"That perfected machines may one day succeed us is, I remember, an extremely commonplace notion on Earth. It prevails not only among poets and romantics but in all classes of society. Perhaps it is because it is so widespread, born spontaneously in popular imagination, that it irritates scientific minds. Perhaps it is also for this very reason that it contains a germ of truth. Only a germ: Machines will always be machines; the most perfected robot, always a robot." (Pierre Boulle, "Planet of the Apes", 1963)

"Science begins with the world we have to live in, accepting its data and trying to explain its laws. From there, it moves toward the imagination: it becomes a mental construct, a model of a possible way of interpreting experience." (Northrop Frye, "The Educated Imagination", 1964)

"The imagination equips us to perceive reality when it is not fully materialized." (Mary C Richards, "Centering in Pottery, Poetry, and the Person", 1964)

"[…] the human reason discovers new relations between things not by deduction, but by that unpredictable blend of speculation and insight […] induction, which - like other forms of imagination - cannot be formalized." (Jacob Bronowski, "The Reach of Imagination", 1967)

"Fantasies are more than substitutes for unpleasant reality; they are also dress rehearsals, plans. All acts performed in the world begin in the imagination." (Barbara G Harrison, [Ms. Magazine] 1973)

"Equilibrium is a figment of the human imagination." (Kenneth Boulding, Toward a General Social Science, 1974)

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On Hypotheses (2000-2009)

"Theoretical physicists are like pure mathematicians, in that they are often interested in the hypothetical behaviour of entirely imaginary objects, such as parallel universes, or particles traveling faster than light, whose actual existence is not being seriously proposed at all." (John Ziman," Real Science: What it Is, and what it Means", 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)

"Given a conjecture, the best thing is to prove it. The second best thing is to disprove it. The third best thing is to prove that it is not possible to disprove it, since it will tell you not to waste your time trying to disprove it. That's what Godel did for the Continuum Hypothesis." (Saharon Shelah, [Rutgers University Colloquium] 2001)

"[Primes] are full of surprises and very mysterious […]. They are like things you can touch […] In mathematics most things are abstract, but I have some feeling that I can touch the primes, as if they are made of a really physical material. To me, the integers as a whole are like physical particles." (Yoichi Motohashi, "The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics", 2002)

"Eliciting and mapping the participant's mental models, while necessary, is far from sufficient [...] the result of the elicitation and mapping process is never more than a set of causal attributions, initial hypotheses about the structure of a system, which must then be tested. Simulation is the only practical way to test these models. The complexity of the cognitive maps produced in an elicitation workshop vastly exceeds our capacity to understand their implications. Qualitative maps are simply too ambiguous and too difficult to simulate mentally to provide much useful information on the adequacy of the model structure or guidance about the future development of the system or the effects of policies." (John D Sterman, "Learning in and about complex systems", Systems Thinking Vol. 3 2003)

"Inaccurate and imprecise measurements or a poor or unrealistic sampling design can result in the generation of inappropriate hypotheses. Measurement errors or a poor experimental design can give a false or misleading outcome that may result in the incorrect retention or rejection of an hypothesis." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"No hypothesis or theory can ever be proven - one day there may be evidence that rejects it and leads to a different explanation (which can include all the successful predictions of the previous hypothesis).Consequently we can only falsify or disprove hypotheses and theories – we can never ever prove them." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"The essential features of the ‘hypothetico-deductive’ view of scientific method are that a person observes or samples the natural world and uses all the information available to make an intuitive, logical guess, called an hypothesis, about how the system functions. The person has no way of knowing if their hypothesis is correct - it may or may not apply. Predictions made from the hypothesis are tested, either by further sampling or by doing experiments. If the results are consistent with the predictions then the hypothesis is retained. If they are not, it is rejected, and a new hypothesis formulated." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"Any physical theory is always provisional, in the sense that it is only a hypothesis: you can never prove it. No matter how many times the results of experiments agree with some theory, you can never be sure that the next time the result will not contradict the theory." (Stephen Hawking, "A Briefer History of Time: The Science Classic Made More Accessible", 2007)

"In mathematics, it’s the limitations of a reasoned argument with the tools you have available, and with magic it’s to use your tools and sleight of hand to bring about a certain effect without the audience knowing what you’re doing. [...]When you’re inventing a trick, it’s always possible to have an elephant walk on stage, and while the elephant is in front of you, sneak something under your coat, but that’s not a good trick. Similarly with mathematical proof, it is always possible to bring out the big guns, but then you lose elegance, or your conclusions aren’t very different from your hypotheses, and it’s not a very interesting theorem." (Persi Diaconis, 2008)

Complexity vs Mathematics II

"[Mathematics] guides our minds in an orderly way, and furnishes us simple and rational principles by means of which ambiguities are clarified, disorder is converted into order, and complexities are analyzed into their component parts." (Johann B Mencken, "The Charlatanry of the Learned", 1715)

"These sciences, Geometry, Theoretical Arithmetic and Algebra, have no principles besides definitions and axioms, and no process of proof but deduction; this process, however, assuming a most remarkable character; and exhibiting a combination of simplicity and complexity, of rigour and generality, quite unparalleled in other subjects." (William Whewell, "The Philosophy of the Inductive Sciences", 1840)

"The value of mathematical instruction as a preparation for those more difficult investigations, consists in the applicability not of its doctrines but of its methods. Mathematics will ever remain the past perfect type of the deductive method in general; and the applications of mathematics to the simpler branches of physics furnish the only school in which philosophers can effectually learn the most difficult and important of their art, the employment of the laws of simpler phenomena for explaining and predicting those of the more complex." (John S Mill, "System of Logic", 1843)

"It is certainly true that all physical phenomena are subject to strictly mathematical conditions, and mathematical processes are unassailable in themselves. The trouble arises from the data employed. Most phenomena are so highly complex that one can never be quite sure that he is dealing with all the factors until the experiment proves it. So that experiment is rather the criterion of mathematical conclusions and must lead the way." (Amos E Dolbear, "Matter, Ether, Motion", 1894)

"Mathematics, the science of the ideal, becomes the means of investigating, understanding and making known the world of the real. The complex is expressed in terms of the simple. From one point of view mathematics may be defined as the science of successive substitutions of simpler concepts for more complex [...]" (William F White, "A Scrap-book of Elementary Mathematics", 1908)

"A great department of thought must have its own inner life, however transcendent may be the importance of its relations to the outside. No department of science, least of all one requiring so high a degree of mental concentration as Mathematics, can be developed entirely, or even mainly, with a view to applications outside its own range. The increased complexity and specialisation of all branches of knowledge makes it true in the present, however it may have been in former times, that important advances in such a department as Mathematics can be expected only from men who are interested in the subject for its own sake, and who, whilst keeping an open mind for suggestions from outside, allow their thought to range freely in those lines of advance which are indicated by the present state of their subject, untrammelled by any preoccupation as to  applications to other departments of science." (Ernst W Hobson, Nature Vol. 84, [address] 1910)

"Elegance may produce the feeling of the unforeseen by the unexpected meeting of objects we are not accustomed to bring together; there again it is fruitful, since it thus unveils for us kinships before unrecognized. It is fruitful even when it results only from the contrast between the simplicity of the means and the complexity of the problem set; it makes us then think of the reason for this contrast and very often makes us see that chance is not the reason; that it is to be found in some unexpected law. In a word, the feeling of  mathematical elegance is only the satisfaction due to any adaptation of the solution to the needs of our mind, and it is because of this very adaptation that this solution can be for us an instrument. Consequently this esthetic satisfaction is bound up with the economy of thought." (Jules Henri Poincaré, "The Future of Mathematics", Monist Vol. 20, 1910)

"The mathematical laws presuppose a very complex elaboration. They are not known exclusively either a priori or a posteriori, but are a creation of the mind; and this creation is not an arbitrary one, but, owing to the mind’s resources, takes place with reference to experience and in view of it. Sometimes the mind starts with intuitions which it freely creates; sometimes, by a process of elimination, it gathers up the axioms it regards as most suitable for producing a harmonious development, one that is both simple and fertile. The mathematics is a voluntary and intelligent adaptation of thought to things, it represents the forms that will allow of qualitative diversity being surmounted, the moulds into which reality must enter in order to become as intelligible as possible." (Émile Boutroux, "Natural Law in Science and Philosophy", 1914)

"No equation, however impressive and complex, can arrive at the truth if the initial assumptions are incorrect." (Arthur C Clarke, "Profiles of the Future: An Inquiry into the Limits of the Possible", 1973)

"Economists are all too often preoccupied with petty mathematical problems of interest only to themselves. This obsession with mathematics is an easy way of acquiring the appearance of scientificity without having to answer the far more complex questions posed by the world we live in." (Thomas Piketty, Capital in the Twenty-First Century, 2013)

On Symbols (1880-1889)

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

"We 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)

"[...] without symbols we could scarcely lift ourselves to conceptual thinking. Thus, in applying the same symbol to different but similar things, we actually no longer symbolize the individual thing, but rather what [the similars] have in common: the concept. This concept is first gained by symbolizing it; for since it is, in itself, imperceptible, it requires a perceptible representative in order to appear to us." (Gottlob Frege, "Über die wissenschaftliche berechtigung einer begriffsschrift", Zeitschrift für Philosophie und philosophische Kritik 81, 1882)

"While all that we have is a relation of phenomena, a mental image, as such, in juxtaposition with or soldered to a sensation, we can not as yet have assertion or denial, a truth or a falsehood. We have mere reality, which is, but does not stand for anything, and which exists, but by no possibility could be true. […] the image is not a symbol or idea. It is itself a fact, or else the facts eject it. The real, as it appears to us in perception, connects the ideal suggestion with itself, or simply expels it from the world of reality. […] you possess explicit symbols all of which are universal and on the other side you have a mind which consists of mere individual impressions and images, grouped by the laws of a mechanical attraction." (Francis H Bradley, "Principles of Logic", 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)

"Pure mathematics proves itself a royal science both through its content and form, which contains within itself the cause of its being and its methods of proof. For in complete independence mathematics creates for itself the object of which it treats, its magnitudes and laws, its formulas and symbols." (Eduard Dillmann, "Die Mathematik die Fackelträgerin einer neuen Zeit", 1889)

On Symbols (1950-1959)

"The most elementary communication is not possible without some degree of conformity to the 'conventions' of the symbolic system." (Talcott Parsons, "The social system", 1951) 

"We could compare mathematics so formalized to a game of chess in which the symbols correspond to the chessmen; the formulae, to definite positions of the men on the board; the axioms, to the initial positions of the chessmen; the directions for drawing conclusions, to the rules of movement; a proof, to a series of moves which leads from the initial position to a definite configuration of the men.” (Friedrich Waismann & Karl Menger, “Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics”, 1951)

"Culture consists of patterns, explicit and implicit, of and for behavior acquired and transmitted by symbols, constituting the distinctive achievement of human groups, including their embodiments in artifacts; the essential core of culture consists of traditional (i.e., historically derived and selected) ideas and especially their attached values; culture systems may, on the one hand, be considered as products of action, on the other as conditioning elements of further action." (Alfred L Kroeber & Clyde Kluckhohn, "Culture", 1952)

"A signal is comprehended if it serves to make us notice the object or situation it bespeaks. A symbol is understood when we conceive the idea it presents." (Susanne Langer, "Feeling and Form: A Theory of Art", 1953)

"Philosophy is in history, and is never independent of historical discourse. But for the tacit symbolism of life it substitutes, in principle, a conscious symbolism; for a latent meaning, one that is manifest. It is never content to accept its historical situation. It changes this situation by revealing it to itself." (Maurice Merleau-Ponty, "Éloge de la philosophie" ["In Praise of Philosophy"], 1953) 

"Beauty had been born, not, as we so often conceive it nowadays, as an ideal of humanity, but as measure, as the reduction of the chaos of appearances to the precision of linear symbols. Symmetry, balance, harmonic division, mated and mensurated intervals – such were its abstract characteristics." (Herbert Read, "Icon and Idea: The Function of Art in the Development of Human Consciousness", 1955)

"A symbol, therefore, may have no effect and indeed ordinarily will have no effect on the image of the immediate future around one. It does produce an effect, however, of what might be called the image of the image, on the image of the future, on the image of the past, on the image of the potential or even of the image of the possible."(Kenneth E Boulding, "The Image: Knowledge in life and society", 1956)

"The symbol and the metaphor are as necessary to science as to poetry." (Jacob Bronowski, "Science and Human Values", 1956)

"To be sure, mathematics can be extended to any branch of knowledge, including economics, provided the concepts are so clearly defined as to permit accurate symbolic representation. That is only another way of saying that in some branches of discourse it is desirable to know what you are talking about." (James R Newman, "The World of Mathematics", 1956)

"Behind these symbols lie the boldest, purest, coolest abstractions mankind has ever made. No schoolman speculating on essences and attributes ever approached anything like the abstractness of algebra." (Susanne K Langer, "Philosophy in a New Key", 1957)

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

"Mathematics is neither a description of nature nor an explanation of its operation; it is not concerned with physical motion or with the metaphysical generation of quantities. It is merely the symbolic logic of possible relations, and as such is concerned with neither approximate nor absolute truth, but only with hypothetical truth. That is, mathematics determines what conclusions will follow logically from given premises. The conjunction of mathematics and philosophy, or of mathematics and science is frequently of great service in suggesting new problems and points of view." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"The mathematician, the statistician, and the philosopher do different things with a theory of probability. The mathematician develops its formal consequences, the statistician applies the work of the mathematician and the philosopher describes in general terms what this application consists in. The mathematician develops symbolic tools without worrying overmuch what the tools are for; the statistician uses them; the philosopher talks about them. Each does his job better if he knows something about the work of the other two." (Irving J Good, "Kinds of Probability", Science Vol. 129 (3347),  1959)

On Symbols (1970-1979)

"Any mathematician endowed with a modicum of intellectual honesty will recognise then that in each of his proofs he is capable of giving a meaning to the symbols he uses." (René Thom, "Modern mathematics, does it exist?", 1972)

"Mathematics in itself, as I say, is independent of experience. It begins with the free choice of symbols, to which are freely assigned properties, and it then proceeds to deduce the necessary rational implications of those properties." (Herbert Dingle, "Science at the Crossroads", 1972)

"A physical theory is assigned a literal and objective interpretation by assigning every one of its referential primitive symbols a physical object - entity, property, relation, or event - rather than a mental picture or a human operation." (Mario Bunge, "Philosophy of Physics", 1973)

"The mind reproduces itself by transmitting its symbols to other intermediaries, human and mechanical, than the particular brain that first assembled them." (Lewis Mumford, "Interpretations and Forecasts 1922-1972", 1973)

"Whether or not a given conceptual model or representation of a physical system happens to be picturable, is irrelevant to the semantics of the theory to which it eventually becomes attached. Picturability is a fortunate psychological occurrence, not a scientific necessity. Few of the models that pass for visual representations are picturable anyhow. For one thing, the model may be and usually is constituted by imperceptible items such as unextended particles and invisible fields. True, a model can be given a graphic representation - but so can any idea as long as symbolic or conventional diagrams are allowed. Diagrams, whether representational or symbolic, are meaningless unless attached to some body of theory. On the other hand theories are in no need of diagrams save for psychological purposes. Let us then keep theoretical models apart from visual analogues."  (Mario Bunge, "Philosophy of Physics", 1973)

"The unconscious reveals its meaning imaginatively, through symbols and images, and it speaks [...] a basically mythological language." (Morton Kelsey, "Myth, History & Faith", 1974)

"Models are not intended to either reflect or construct a single objective reality. Rather, their purpose is to simulate some aspect of a possible reality. In NLP, for instance, it is not important whether or not a model is 'true' , but rather that it is 'useful' . In fact, all models can be perceived as symbolic or metaphoric, as opposed to reflective of reality. Whether the description being used is metaphorical or literal, the usefulness of a model depends on the degree to which it allows us to move effectively to the next step in the sequence of transformations connecting deeper structures and surface structures. Instead of 'constructing' reality, models establish a set of functions that serve as a tool or a bridge between deep structures and surface structures. It is this bridge that forms our 'understanding' of reality and allows us to generate new experiences and expressions of reality." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"Symbols, formulae and proofs have another hypnotic effect. Because they are not immediately understood, they, like certain jokes, are suspected of holding in some sort of magic embrace the secret of the universe, or at least some of its more hidden parts." (Scott Buchanan, "Poetry and Mathematics", 1975)

"Symbol and myth do bring into awareness infantile, archaic dreads and similar primitive psychic content. This is their regressive aspect. But they also bring out new meaning, new forms, and disclose a reality that was literally not present before, a reality that is not merely subjective but has a second pole which is outside ourselves. This is the progressive side of symbol and myth." (Rollo May, "The Courage to Create", 1975) 

"The most pervasive paradox of the human condition which we see is that the processes which allow us to survive, grow, change, and experience joy are the same processes which allow us to maintain an impoverished model of the world - our ability to manipulate symbols, that is, to create models. So the processes which allow us to accomplish the most extraordinary and unique human activities are the same processes which block our further growth if we commit the error of mistaking the model of the world for reality." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"Whenever the Eastern mystics express their knowledge in words - be it with the help of myths, symbols, poetic images or paradoxical statements-they are well aware of the limitations imposed by language and 'linear' thinking. Modern physics has come to take exactly the same attitude with regard to its verbal models and theories. They, too, are only approximate and necessarily inaccurate. They are the counterparts of the Eastern myths, symbols and poetic images, and it is at this level that I shall draw the parallels. The same idea about matter is conveyed, for example, to the Hindu by the cosmic dance of the god Shiva as to the physicist by certain aspects of quantum field theory. Both the dancing god and the physical theory are creations of the mind: models to describe their authors' intuition of reality." (Fritjof Capra, "The Tao of Physics: An Exploration of the Parallels Between Modern Physics and Eastern Mysticism", 1975)

"The chief difficulty of modern theoretical physics resides not in the fact that it expresses itself almost exclusively in mathematical symbols, but in the psychological difficulty of supposing that complete nonsense can be seriously promulgated and transmitted by persons who have sufficient intelligence of some kind to perform operations in differential and integral calculus […]" (Celia Green, "The Decline and Fall of Science", 1976)

"[…] all our symbols have the same purpose; words are merely the symbols we use most commonly. The function of words in human thought is to stand for things which are not present to the senses, and allow the mind to manipulate them - things, concepts, ideas, everything that does not have a physical reality in front of us now." (Jacob Bronowski, "The Imaginative Mind in Art", 1978) 

"[Human consciousness] depends wholly on our seeing the outside world in such categories. And the problems of consciousness arise from putting reconstitution beside internalization, from our also being able to see ourselves as if we were objects in the outside world. That is in the very nature of language; it is impossible to have a symbolic system without it." (Jacob Bronowski, "The origins of knowledge and imagination", 1978)

"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)