"Without the sensuous faculty no object would be given to us, without understanding no object would be thought. Thoughts without content are void, intuitions without conceptions, blind." Immanuel Kant, "Critique of Pure Reason", 1781)
"The discovery of truth by slow, progressive meditation is talent. Intuition of the truth, not preceded by perceptible meditation, is genius." (Johann K Lavater, 1787)
"If the human mind is nonetheless to be able even to think the given infinite without contradiction, it must have within itself a power that is supersensible, whose idea of the noumenon cannot be intuited but can yet be regarded as the substrate underlying what is mere appearance, namely, our intuition of the world [worldview]." (Immanuel Kant, "Critique of Judgment", 1790)
"We construct concepts when we represent them in intuition a priori, without experience, or when we represent in intuition the object which corresponds to our concept of it. - The mathematician can never apply his reason to mere concepts, nor the philosopher to the construction of concepts. - In mathematics the reason is employed in concreto, however, the intuition is not empirical, but the object of contemplation is something a priori." (Immanuel Kant, "Logic", 1800)
"There are two things cheap and common enough when separated, but as costly in value, as irresistible in power, when combined - truth and novelty. Their union is like that of steam and of fire, which nothing can overcome. Truth and novelty, when united, must overthrow the whole superincumbent pressure of error and of prejudice, whatever be its weight; and the effects will be proportionate to the resistance. But the moral earthquake, unlike the natural, while it convulses the nations, reforms them too." (Charles C Colton, "Lacon", 1820)
"This is the test and triumph of originality, not to show us what has never been, and what we may therefore very easily never have dreamt of, but to point out to us what is before our eyes and under our feet, though we have had no suspicion of its existence, for want of sufficient strength of intuition, of determined grasp of mind to seize and retain it." (William Hazlitt, "Table Talk; or, Original Essays", 1821)
"[...] it should be noted that the seeds of wisdom that are to bear fruit in the intellect are sown less by critical studies and learned monographs than by insights, broad impressions, and flashes of intuition." (Carl von Clausewitz, "On War", 1832)
"Logic is the procession or proportionate unfolding of the intuition; but its virtue is as silent method; the moment it would appear as propositions, and have a separate value, it is worthless." (Ralph W Emerson, "Essays", 1841)
"We study the complex in the simple; and only from the intuition of the lower can we safely proceed to the intellection of the higher degrees. The only danger lies in the leaping from low to high, with the neglect of the intervening gradations." (Samuel T Coleridge, "Physiology of Life", 1848)
"Truths are known to us in two ways: some are known directly, and of themselves; some through the medium of other truths. The former are the subject of Intuition, or Consciousness; the latter, of Inference; the latter of Inference. The truths known by Intuition are the original premises, from which all others are inferred." (John S Mill, "A System of Logic, Ratiocinative and Inductive", 1858)
"The man who is guided by concepts and abstractions only succeeds by such means in warding off misfortune, without ever gaining any happiness for himself from these abstractions. And while he aims for the greatest possible freedom from pain, the intuitive man, standing in the midst of a culture, already reaps from his intuition a harvest of continually inflowing illumination, cheer, and redemption - in addition to obtaining a defense against misfortune. To be sure, he suffers more intensely, when he suffers; he even suffers more frequently, since he does not understand how to learn from experience and keeps falling over and over again into the same ditch." (Friedrich Nietzsche," On Truth and Lie in an Extra-Moral Sense", 1873)
"With our notion of the essence of intuition, an intuitive treatment of figurative representations will tend to yield a certain general guide on which mathematical laws apply and how their general proof may be structured. However, true proof will only be obtained if the given figures are replaced with figures generated by laws based on the axioms and these are then taken to carry through the general train of thought in an explicit case. Dealing with sensate objects gives the mathematician an impetus and an idea of the problems to be tackled, but it does not pre-empt the mathematical process itself. (Felix Klein, "Nicht-Euklidische Geometrie I: Vorlesung gehalten während des Wintersemesters 1889–90", 1892)
" […] the naive intuition is not exact, while the refined intuition is not properly intuition at all, but arises through the logical development from axioms considered as perfectly exact." (Felix Klein, [lectures] 1893)
"It is not easy to anatomize the constitution and the operations of a mind which makes such an advance in knowledge. Yet we may observe that there must exist in it, in an eminent degree, the elements which compose the mathematical talent. It must possess distinctness of intuition, tenacity and facility in tracing logical connection, fertility of invention, and a strong tendency to generalization." (William Whewell, "History of the Inductive Sciences" Vol. 1, 1894)
"The scientific value of truth is not, however, ultimate or absolute. It rests partly on practical, partly on aesthetic interests. As our ideas are gradually brought into conformity with the facts by the painful process of selection, - for intuition runs equally into truth and into error, and can settle nothing if not controlled by experience, - we gain vastly in our command over our environment. This is the fundamental value of natural science" (George Santayana, "The Sense of Beauty: Being the Outlines of Aesthetic Theory", 1896)
"Incidentally, naive intuition, which is in large part an inherited talent, emerges unconsciously from the in-depth study of this or that field of science. The word ‘Anschauung’ has not perhaps been suitably chosen. I would like to include here the motoric sensation with which an engineer assesses the distribution of forces in something he is designing, and even that vague feeling possessed by the experienced number cruncher about the convergence of infinite processes with which he is confronted. I am saying that, in its fields of application, mathematical intuition understood in this way rushes ahead of logical thinking and in each moment has a wider scope than the latter " (Felix Klein, "Über Arithmetisierung der Mathematik", Zeitschrift für mathematischen und naturwissen-schaftlichen Unterricht 27, 1896)
"New ideas emerge dimly into intuition, come into consciousness from nobody knows where, and become the material on which the mind operates, forging them gradually into consistent doctrine, which can be welded on to existing domains of knowledge. But this process is never complete: a crude connection can always be pointed to by a logician as an indication of the imperfection of human constructions." (Henri Poincaré, "Science and Hypothesis", 1902)
"The chief end of mathematical instruction is to develop certain powers of the mind, and among these the intuition is not the least precious. By it the mathematical world comes in contact with the real world, and even if pure mathematics could do without it, it would always be necessary to turn to it to bridge the gulf between symbol and reality. The practician will always need it, and for one mathematician there are a hundred practicians. However, for the mathematician himself the power is necessary, for while we demonstrate by logic, we create by intuition; and we have more to do than to criticize others’ theorems, we must invent new ones, this art, intuition teaches us." (Jules H Poincaré, "The Value of Science", 1905)
"We believe that in our reasonings we no longer appeal to intuition; the philosophers will tell us this is an illusion. Pure logic could never lead us to anything but tautologies; it could create nothing new; not from it alone can any science issue. In one sense these philosophers are right; to make arithmetic, as to make geometry, or to make any science, something else than pure logic is necessary. To designate this something else we have no word other than intuition. But how many different ideas are hidden under this same word?" (Henri Poincaré , "Intuition and Logic in Mathematics", 1905)
"It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better." (Henri Poincaré, "Science and Method", 1908)
"Mathematics is merely a shorthand method of recording physical intuition and physical reasoning, but it should not be a formalism leading from nowhere to nowhere, as it is likely to be made by one who does not realize its purpose as a tool." (Charles P Steinmetz, "Transactions of the American Institute of Electrical Engineers", 1909)
"Pure mathematics is a collection of hypothetical, deductive theories, each consisting of a definite system of primitive, undefined, concepts or symbols and primitive, unproved, but self-consistent assumptions (commonly called axioms) together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition." (Graham Fitch, "The Fourth Dimension simply Explained", 1910)
"Knowledge is the distilled essence of our intuitions, corroborated by experience." (Elbert Hubbard, "A Thousand & One Epigrams, 1911)
"In fact, the opposition of instinct and reason is mainly illusory. Instinct, intuition, or insight is what first leads to the beliefs which subsequent reason confirms or confutes; but the confirmation, where it is possible, consists, in the last analysis, of agreement with other beliefs no less instinctive. Reason is a harmonising, controlling force rather than a creative one. Even in the most purely logical realms, it is insight that first arrives at what is new." (Bertrand Russell, "Our Knowledge of the External World", 1914)
"The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them."(Albert Einstein, "Principles of Research", 1918)
"We must have a logical intuition of the probable relations between propositions. Once the existence of this relation between evidence and conclusion, the latter becomes the subject of the degree of belief." (John M Keynes, "Treatise on Probability", 1921)
"By intuition is frequently understood perception, or the knowledge of actual reality, the apprehension of something as real. […] Intuition is the undifferentiated unity of the perception of the real and of the simple image of the possible. " (Benedetto Croce, "The Essence of Æsthetic", 1921)
"Intuition does not denote something contrary to reason, but something outside of the province of reason." (Carl Gustav Jung, "Psychological types: or, The psychology of individuation", 1926)
"Scientific hypotheses are intuitive leaps in the dark." (Alexander Goldenweiser, "Robots or Gods: An Essay on Craft and Mind", American Journal of Sociology 37 (3), 1931)
"There is no such thing as a logical method of having new ideas or a logical reconstruction of this process […] very discovery contains an ‘irrational element’ or a ‘creative intuition’." (Karl Popper, "The logic of scientific discover", 1934)
"Men of science belong to two different types - the logical and the intuitive. Science owes its progress to both forms of minds. Mathematics, although a purely logical structure, nevertheless makes use of intuition. " (Alexis Carrel, "Man the Unknown", 1935)
"Science does not mean an idle resting upon a body of certain knowledge; it means unresting endeavor and continually progressing development toward an end which the poetic intuition may apprehend, but which the intellect can never fully grasp." (Max Planck, "The Philosophy of Physics", 1936)
"When a scientist is ahead of his times, it is often through misunderstanding of current, rather than intuition of future truth. In science there is never any error so gross that it won't one day, from some perspective, appear prophetic." (Jean Rostand, "Pensées d'un Biologiste",1939)
"The intellect is at home in that which is fixed only because it is done and over with, for intellect is itself just as much a deposit of past life as is the matter to which it is congenial. Intuition alone articulates in the forward thrust of life and alone lays hold of reality." (John Dewey, "Time and Individuality", 1940)
"It is his intuition, his mystical insight into the nature of things, rather than his reasoning which makes a great scientist." (Karl R Popper, "The Open Society and Its Enemies", 1945)
"But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perception will agree with them and, moreover, to believe that a question not decidable now has meaning and may be decided in future." (Kurt Gödel, "What is Cantor’s Continuum problem?", American Mathematical Monthly 54, 1947)
"[...] when the pioneer in science sends for the groping feelers of his thoughts, he must have a vivid intuitive imagination, for new ideas are not generated by deduction, but by an artistically creative imagination. Nevertheless, the worth of a new idea is invariably determined, not by the degree of its intuitiveness - which, incidentally, is to a major extent a matter of experience and habit - but by the scope and accuracy of the individual laws to the discovery of which it eventually leads. (Max Planck, "The Meaning and Limits of Exact Science", Science Vol. 110 (2857), 1949)
"[…] 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)
"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. 3 (11),1953)
"Mathematicians create by acts of insight and intuition. Logic then sanctions the conquests of intuition. It is the hygiene that mathematics practice to keep its ideas healthy and strong. Moreover, the whole structure rests fundamentally on uncertain ground, the intuitions of man." (Morris Kline, "Mathematics in Western Culture", 1953)
"The construction of hypotheses is a creative act of inspiration, intuition, invention; its essence is the vision of something new in familiar material." (Milton Friedman, "Essays in Positive Economics", 1953)
"Science, then, is the attentive consideration of common experience; it is common knowledge extended and refined. Its validity is of the same order as that of ordinary perception; memory, and understanding. Its test is found, like theirs, in actual intuition, which sometimes consists in perception and sometimes in intent." (George Santayana, "The Life of Reason, or the Phases of Human Progress", 1954)
"The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them." (Albert Einstein,"Ideas and Opinions", 1954)
"There comes a point where the mind takes a leap - call it intuition or what you will - and comes out upon a higher plane of knowledge, but can never prove how it got there. All great discoveries have involved such a leap." (Albert Einstein, [interview in Life, "Death of a Genius"] 1955)
"Science does not mean an idle resting upon a body of certain knowledge; it means unresting endeavor and continually progressing development toward an end which the poetic intuition may apprehend, but which the intellect can never fully grasp." (Max Planck, "The New Science", 1959)
"Intuition implies the act of grasping the meaning or significance or structure of a problem without explicit reliance on the analytical apparatus of one’s craft. It is the intuitive mode that yields hypotheses quickly, that produces interesting combinations of ideas before their worth is known. It precedes proof: indeed, it is what the techniques of analysis and proof are designed to test and check. It is founded on a kind of combinatorial playfulness that is only possible when the consequences of error are not overpowering or sinful." (Jerome S Bruner, "On Learning Mathematics", Mathematics Teacher Vol. 53, 1960)
"The functional validity of a working hypothesis is not a priori certain, because often it is initially based on intuition. However, logical deductions from such a hypothesis provide expectations (so called prognoses) as to the circumstances under which certain phenomena will appear in nature. Such a postulate or working hypothesis can then be substantiated by additional observations or by experiments especially arranged to test details. The value of the hypothesis is strengthened if the observed facts fit the expectation within the limits of permissible error." (R Willem van Bemmelen, "The Scientific Character of Geology", The Journal of Geology Vol 69 (4), 1961)
"Intuition is the collection of odds and ends where we place all the intellectual mechanisms which we do not know how to analyze or even name with precision, or which we are not interested in analyzing or naming." (Mario Bunge, "Intuition and Science", 1962)
"The moment of truth, the sudden emergence of new insight, is an act of intuition. Such intuitions give the appearance of miraculous flashes, or short circuits of reasoning. In fact they may be likened to an immersed chain, of which only the beginning and the end are visible above the surface of consciousness. The diver vanishes at one end of the chain and comes up at the other end, guided by invisible links." (Arthur Koestler, "The Act of Creation", 1964)
"People may come along and argue philosophically that they like one better than another; but we have learned from much experience that all philosophical intuitions about what nature is going to do fail." (Richard Feynman, "The Character of Physical Law", 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)
"Scientific research was much like prospecting: you went out and you hunted, armed with your maps and instruments, but in the end your preparations did not matter, or even your intuition. You needed your luck, and whatever benefits accrued to the diligent, through sheer, grinding hard work." (Michael Crichton, "The Andromeda Strain", 1969)
"The voice from within can occasionally be wrong, even in the wisest individual, in any case, such wise individuals generally test their inner commands against external reality whenever they can. Empirical testing and verifying of experiential knowledge is thus always in order, for sometimes the inner certainty, even of a veritable mystic, turns out to be the voice of the devil. It is not yet wise to permit the private conscience of one person to outweigh, all other sources of knowledge and wisdom, however much we respect inner experiencing." (Abraham Maslow, "The FFartherReaches of Human Nature", 1971)
"Science does not need mysticism and mysticism does not need science, but man needs both. Mystical experience is necessary to understand the deepest nature of things, and science is essential for modern life. What we need, therefore, is not a synthesis, but a dynamic interplay between mystical intuition and scientific analysis." (Fritjof Capra, "The Tao of Physics: An Exploration of the Parallels Between Modern Physics and Eastern Mysticism", 1975)
"[…] the distinction between rigorous thinking and more vague ‘imaginings’; even in mathematics itself, all is not a question of rigor, but rather, at the start, of reasoned intuition and imagination, and, also, repeated guessing. After all, most thinking is a synthesis or juxtaposition of advances along a line of syllogisms - perhaps in a continuous and persistent ‘forward'’ movement, with searching, so to speak ‘sideways’, in directions which are not necessarily present from the very beginning and which I describe as ‘sending out exploratory patrols’ and trying alternative routes." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)
"Real progress in understanding nature is rarely incremental. All important advances are sudden intuitions, new principles, new ways of seeing." (Marilyn Ferguson, "The Aquarian Conspiracy: Personal and Social Transformation in the 1980s", 1980)
"We are now compelled to accept the fact that there is no such thing as an absolute proof or a universally acceptable proof. We know that, if we question the statements we accept on an intuitive basis, we shall be able to prove them only if we accept others on an intuitive basis." (Morris Kline, "Mathematics: The loss of certainty", 1980)
"[…] science must be understood as a social phenomenon, a gutsy, human enterprise, not the work of robots programmed to collect pure information. […] Science, since people must do it, is a socially embedded activity. It progresses by hunch, vision, and intuition." (Stephen J Gould, "The Mismeasure of Man", 1980)
"That is to say, intuition is not a direct perception of something existing externally and eternally. It is the effect in the mind of certain experiences of activity and manipulation of concrete objects (at a later stage, of marks on paper or even mental images)." (Philip J Davis & Reuben Hersh, "The Mathematical Experience", 1981)
"Analytic thought is based on detailed defined relations between two elements at a time. Intuitive thought is based on an emotional state associated with all the elements in the field of knowledge (overall impression). " (Tony Bastick, "Intuition: How we think and act", 1982)
"Intuition takes many different forms in science and mathematics, though all forms of it have certain properties in common: the suddenness of their origin, the wholeness of the conception they embody, and the absence of conscious premeditation." (Peter Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)
"Intuition can be understood as the mind turning in on itself and apprehending the result of processes that have taken place outside of awareness." (Philip Goldberg, "The Intuitive Edge", 1983)
"It seems that truth is progressive approximation in which the relative fraction of our spontaneously tolerated residual error constantly diminishes." (R Buckminster Fuller, "Humans in Universe", 1983)
"The assumptions and definitions of mathematics and science come from our intuition, which is based ultimately on experience. They then get shaped by further experience in using them and are occasionally revised. They are not fixed for all eternity." (Richard Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"The common perception of science as a rational activity, in which one confronts the evidence of fact with an open mind, could not be more false. Facts assume significance only within a pre-existing intellectual structure, which may be based as much on intuition and prejudice as on reason." (Walter Gratzer, The Guardian, 1989)
"Every theoretical explanation is a reduction of intuition." (Peter Høeg, "Smilla's Sense of Snow", 1992)
"Intuition is the art, peculiar to the human mind, of working out the correct answer from data that is, in itself, incomplete or even, perhaps, misleading." (Isaac Asimov, "Forward the Foundation", 1993)
"All this points to an appealing intuition: that a faculty for analogical reasoning is an innate part of human cognition." (Dedre Gentner & Michael Jeziorski, "Western science", 1993)
"Many pages have been expended on polemics in favor of rigor over intuition, or of intuition over rigor. Both extremes miss the point: the power of mathematics lies precisely in the combination of intuition and rigor." (Ian Stewart, "Concepts of Modern Mathematics", 1995)
"Scientists reach their conclusions for the damnedest of reasons: intuition, guesses, redirections after wild-goose chases, all combing with a dollop of rigorous observation and logical reasoning to be sure […] This messy and personal side of science should not be disparaged, or covered up, by scientists for two major reasons. First, scientists should proudly show this human face to display their kinship with all other modes of creative human thought […] Second, while biases and references often impede understanding, these mental idiosyncrasies may also serve as powerful, if quirky and personal, guides to solutions." (Stephen J Gould, "Dinosaur in a Haystack: Reflections in natural history", 1995)
"Intuition isn't direct perception of something external. It's the effect in the mind/brain of manipulating concrete objects - at a later stage, of making marks on paper, and still later, manipulating mental images. This experience leaves a trace, an effect, in your mind/brain." (Reuben Hersh, "What Is Mathematics, Really?", 1998)
"All perceiving is also thinking, all reasoning is also intuition, all observation is also invention." (Rudolf Arnheim, "Art and Visual Perception: A Psychology of the Creative Eye", 2004)
"Patterns experienced again and again become intuitions. […] Intuitive judgments are made by our use of imagery; intuition is the result of mental model building. […] The mental model used and the form of the intuition is dependent upon the question being answered." (Roger Frantz,"Two Minds", 2005)
"This covert mechanism would be the source of what we call intuition, the mysterious mechanism by which we arrive at the solution of a problem without reasoning toward it." (Antonio Damasio, "Descartes' Error", 2005)
"It is from this continuousness of thought and perception that the scientist, like the writer, receives the crucial flash of insight out of which a piece of work is conceived and executed. And the scientist (again like the writer) is grateful when the insight comes, because insight is the necessary catalyst through which the abstract is made concrete, intuition be given language, language provides specificity, and real work can go forward." (Vivian Gornick, "Women in Science: Then and Now", 2009)
"Without denying the usefulness of the distinction between intuition and proof, I believe it can be drawn too sharply; intuition plays an essential role in the making and evaluating of proofs and is sometimes changed as a consequence of these processes. In this respect, the distinction is like that between creative and critical thinking; while this distinction too is a useful one, it is not possible to have either in any very satisfactory sense without the other." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2010)
"An intuition is neither caprice nor a sixth sense but a form of unconscious intelligence." (Gerd Gigerenzer, "Risk Savvy", 2015)
"That is the trouble with facts: they sometimes force you to conclusions that differ with your intuition." (Steven G Krantz, "A Primer of Mathematical Writing" 2nd Ed., 2016)
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