Carl Friedrich Gauss - Collected Quotes

"[…] the way in which I have proceeded does not lead to the desired goal, the goal that you declare you have reached, but instead to a doubt of the validity of [Euclidean] geometry. I have certainly achieved results which most people would look upon as proof, but which in my eyes prove almost nothing; if, for example, one can prove that there exists a right triangle whose area is greater than any given number, then I am able to establish the entire system of [Euclidean] geometry with complete rigor. Most people would certainly set forth this theorem as an axiom; I do not do so, though certainly it may be possible that, no matter how far apart one chooses the vertices of a triangle, the triangle's area still stays within a finite bound. I am in possession of several theorems of this sort, but none of them satisfy me." (Carl F Gauss, 1799) [answer to a letter from Farkas Bolyai in which Bolyai claimed to have proved Euclid's fifth postulate]

“Number theory is revealed in its entire simplicity and natural beauty when the field of arithmetic is extended to the imaginary numbers” (Carl F Gauss, “Disquisitiones arithmeticae” [“Arithmetical Researches”], 1801)

"The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers. […] Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated." (Carl F Gauss, "Disquisitiones Arithmeticae” [“Arithmetical Researches”], 1801)

"It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never satisfied man is so strange if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretched out his arms for others." (Carl F Gauss, [Letter to Farkas Bolyai] 1808)

"At the beginning I would ask anyone who wants to introduce a new function in analysis to clarify whether he intends to confine it to real magnitudes (real values of the argument) and regard the imaginary values as just vestigial –or whether he subscribes to my fundamental proposition that in the realm of magnitudes the imaginary ones a+b√−1 = a+bi have to be regarded as enjoying equal rights with the real ones. We are not talking about practical utility here; rather analysis is, to my mind, a self-sufficient science. It would lose immeasurably in beauty and symmetry from the rejection of any fictive magnitudes. At each stage truths, which otherwise are quite generally valid, would have to be encumbered with all sorts of qualifications." (Carl F Gauss, [letter to Bessel] 1811)

"What should one understand by ∫ ϕx · dx for x = a + bi? Obviously, if we want to start from clear concepts, we have to assume that x passes from the value for which the integral has to be 0 to x = a + bi through infinitely small increments (each of the form x = a + bi), and then to sum all the ϕx · dx. Thereby the meaning is completely determined. However, the passage can take placein infinitely many ways: Just like the realm of all real magnitudes can be conceived as an infinite straight line, so can the realm of all magnitudes, real and imaginary, be made meaningful by an infinite plane, in which every point, determined by abscissa = a and ordinate = b, represents the quantity a+bi. The continuous passage from one value of x to another a+bi then happens along a curve and is therefore possible in infinitely many ways. I claim now that after two different passages the integral ∫ ϕx · dx acquires the same value when ϕx never becomes equal to ∞ in the region enclosed by the two curves representing the two passages." (Carl F Gauss, [letter to Bessel] 1811)

"I am convinced more and more that the necessary truth of our geometry cannot be demonstrated, at least not by the human intellect to the human understanding. Perhaps in another world we may gain other insights into the nature of space which at present are unattainable to us. Until then we must consider geometry as of equal rank not with arithmetic, which is purely a priori, but with mechanics." (Carl Friedrich Gauss, [Letter to Olbers] 1817)

“It is characteristic of higher arithmetic that many of its most beautiful theorems can be discovered by induction with the greatest of ease but have proofs that lie anywhere but near at hand and are often found only after many fruitless investigations with the aid of deep analysis and lucky combinations.” (Carl F Gauss, 1817)

"From the foregoing we see that the two justifications each leave something to be desired. The first depends entirely on the hypothetical form of the probability of the error; as soon as that form is rejected, the values of the unknowns produced by the method of least squares are no more the most probable values than is the arithmetic mean in the simplest case mentioned above. The second justification leaves us entirely in the dark about what to do when the number of observations is not large. In this case the method of least squares no longer has the status of a law ordained by the probability calculus but has only the simplicity of the operations it entails to recommend it." (Carl Friedrich Gauss, "Anzeige: Theoria combinationis observationum erroribus minimis obnoxiae: Pars prior", 

Göttingische gelehrte Anzeigen, 1821)

"We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori." (Carl F Gauss, 1830)

"[…] if number is merely the product of our mind, space has a reality outside our mind whose laws we cannot a priori completely prescribe" (Carl F Gauss, 1830)

"[geometrical representation of complex numbers] completely established the intuitive meaning of complex numbers, and more is not needed to admit these quantities into the domain of arithmetic." (Carl F Gauss, 1831)

“I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a facon de parler [manner of speaking], the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without restriction.” (Carl F Gauss, 1831)

"Complete knowledge of the nature of an analytic function must also include insight into its behavior for imaginary values of the arguments. Often the latter is indispensable even for a proper appreciation of the behavior of the function for real arguments. It is therefore essential that the original determination of the function concept be broadened to a domain of magnitudes which includes both the real and the imaginary quantities, on an equal footing, under the single designation complex numbers." (Carl F Gauss, cca. 1831)

"The Higher Arithmetic presents us with an inexhaustible storehouse of interesting truths - of truths, too, which are not isolated but stand in the closest relation to one another, and between which, with each successive advance of the science, we continually discover new and sometimes wholly unexpected points of contact. A great part of the theories of Arithmetic derive an additional charm from the peculiarity that we easily arrive by induction at important propositions which have the stamp of simplicity upon them but the demonstration of which lies so deep as not to be discovered until after many fruitless efforts; and even then it is obtained by some tedious and artificial process while the simpler methods of proof long remain hidden from us." (Carl F Gauss, [introduction to Gotthold Eisenstein’s "Mathematische Abhandlungen"] 1847)

"I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible." (Carl F Gauss)

“In the Theory of Numbers it happens rather frequently that, by some unexpected luck, the most elegant new truths spring up by induction.” (Carl F Gauss)

"That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question." (Carl F Gauss)

René Descartes - Collected Quotes

"Divide each problem that you examine into as many parts as you can and as you need to solve them more easily." (René Descartes, “Discourse on Method”, 1637)

"Each problem that I solved became a rule which served afterwards to solve other problems. [...] thus each truth discovered was a rule available in the discovery of subsequent ones.” (René Descartes, “Discourse on Method”, 1637)

“[…] neither the true roots nor the false are always real; sometimes they are, however, imaginary; namely, whereas we can always imagine as many roots for each equation as I have predicted, there is still not always a quantity which corresponds to each root so imagined. Thus, while we may think of the equation x^3 - 6xx + 13x - 10 = 0 as having three roots, yet there is just one real root, which is 2, and the other two [2+i and 2-i]], however, increased, diminished, or multiplied them as we just laid down, remain always imaginary.” (René Descartes, “Gemetry”, 1637)

"The long chains of simple and easy reasonings by means of which geometers are accustomed to reach the conclusions of their most difficult demonstrations, had led me to imagine that all things, to the knowledge of which man is competent, are mutually connected in the same way, and that there is nothing so far removed from us as to be beyond our reach, or so hidden that we cannot discover it, provided only we abstain from accepting the false for the true, and always preserve in our thoughts the order necessary for the deduction of one truth from another." (René Descartes, "Discourse on the Method", 1637)

“From all this I am beginning to have a rather better understanding of what I am. But it still appears - and I cannot stop thinking this - that the corporeal things of which images are formed in my thought, and which the senses investigate, are known with much more distinctness than this puzzling 'I' which cannot be pictured in the imagination.” (René Descartes, “Meditations” II, 1641)

“In order to seek truth, it is necessary once in the course of our life, to doubt, as far as possible, of all things.” (René Descartes, “Principles of Philosophy”, 1644)

"There is nothing in us which we must attribute to our soul except our thoughts. These are of two principal kinds, some being actions of the soul and others its passions. Those I call its actions are all our volitions […] the various perceptions or modes of knowledge present in us may be called its passions in the general sense, for it is often not our soul which makes them such as they are, and the soul always receives them from the things that are represented by them." (René Descartes, "Les passions de l’âme" ["Passions of the Soul"], 1649)

"The two operations of our understanding [… are] intuition and deduction, on which alone we have said we must rely in the acquisition of knowledge." (René Descartes, "Rules for the Direction of the Mind", 1684)

"When a problem arises, we should be able to see soon whether it will be profitable to examine some other problems first, and which others, and in which order." (René Descartes, "OEuvres" Vol. X, 1897)

"All philosophy is like a tree, the roots are metaphysics, the trunk is physics, and the branches are all the other sciences.” (René Descartes)

"Intuition is the conception of an attentive mind, so clear, so distinct, and so effortless that we cannot doubt what we have so conceived.” (René Descartes)

“It is very helpful to represent these things in this fashion since nothing enters the mind more readily than geometric figures.” (René Descartes)

"The two operations of our understanding, intuition and deduction, on which alone we have said we must rely in the acquisition of knowledge." (René Descartes)

"There have been only Mathematicians who were able to find some proofs, that is to say some sure and certain reasons." (René Descartes)

“We call infinite that thing whose limits we have not perceived, and so by that word we do not signify what we understand about a thing, but rather what we do not understand.” (René Descartes)

Albert Einstein - Collected Quotes

"As soon as science has emerged from its initial stages, theoretical advances are no longer achieved merely by a process of arrangement. Guided by empirical data, the investigator rather develops a system of thought which, in general, is built up logically from a small number of fundamental assumptions, the so-called axioms. We call such a system of thought a theory. The theory finds the justification for its existence in the fact that it correlates a large number of single observations, and it is just here that the 'truth' of the theory lies.  " (Albert Einstein, "Relativity: The Special and General Theory ", 1916)

"No fairer destiny could be allotted to any physical theory, than that it should of itself point out the way to the introduction of a more comprehensive theory, in which it lives on as a limiting case. " (Albert Einstein, "Relativity, The Special and General Theory ", 1916)

"Since the introduction of the special principle of relativity has been justified, every intellect which strives after generalization must feel the temptation to venture the step towards the general principle of relativity." (Albert Einstein, 1917)

"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)

"Most teachers waste their time by asking questions which are intended to discover what a pupil does not know whereas the true art of questioning has for its purpose to discover what the pupil knows or is capable of knowing." (Albert Einstein, 1920)

"The discovery of Minkowski […] is to be found […] in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude, √-1*ct, proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space-coordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry." (Albert Einstein,"Relativity: The Special and General Theory", 1920)

"A geometrical-physical theory as such is incapable of being directly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind. To ‘visualise’ a theory, or bring it home to one's mind, therefore means to give a representation to that abundance of experiences for which the theory supplies the schematic arrangement " (Albert Einstein,  "Geometry and Experience ", 1921)

"Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch of physics. Its affirmations rest essentially on induction from experience, but not on logical inferences only. We call this 'practical geometry'. [...] The question whether the practical geometry of the universe is Euclidean or not has a clear meaning, and its answer can only be furnished by experience." (Albert Einstein, [lecture] 1921)

"It seems that the human mind has first to construct forms independently, before we can find them in things. Kepler’s marvelous achievement is a particularly fine example of the truth that knowledge cannot spring from experience alone, but only from the comparison of the inventions of the intellect with observed fact." (Albert Einstein, 1930)

"Physics is the attempt at the conceptual construction of a model of the real world and its lawful structure. " (Albert Einstein, [letter to Moritz Schlick] 1931)

"It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience. " (Albert Einstein, [lecture] 1933)

"Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships.  In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature. " (Albert Einstein, [Obituary for Emmy Noether], 1935)

"Creating a new theory is not like destroying an old barn and erecting a skyscraper in its place. It is rather like climbing a mountain, gaining new and wider views, discovering unexpected connections between our starting point and its rich environment. But the point from which we started out still exists and can be seen, although it appears smaller and forms a tiny part of our broad view gained by the mastery of the obstacles on our adventurous way up. " (Albert Einstein & Leopold Infeld,  "The Evolution of Physics ", 1938)

"Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone."  (Albert Einstein & Leopold Infeld,  "The Evolution of Physics", 1938)

"Physical concepts are free creations of the human mind, and are not, however it may seem, uniquely determined by the external world In our endeavor to understand reality we are somewhat like a man trying to understand the mechanism of a closed watch. He sees the face and the moving hands, even hears its ticking, but he has no way of opening the case. If he is ingenious he may form some picture of a mechanism which could be responsible for all the things he observes, but he may never be quite sure his picture is the only one which could explain his observations. He will never be able to compare his picture with the real mechanism and he cannot even imagine the possibility of the meaning of such a comparison." (Albert Einstein & Leopold Infeld,  "The Evolution of Physics ", 1938)

"The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and marks real advances in science." (Albert Einstein & Leopold Infeld, "The Evolution of Physics", 1938)

"With the help of physical theories we try to find our way through the maze of observed facts, to order and understand the world of our sense impressions. " (Albert Einstein & Leopold Infeld,  "The Evolution of Physics ", 1938)

"The development of science and of creative activities of the spirit in general requires still another kind of freedom, which may be characterized as inward freedom. It is this freedom of the spirit which consists in the independence of thought from the restrictions of authoritarian and social prejudices as well as from unphilosophical routinizing and habit in general." (Albert Einstein, "On Freedom", 1940)

"When the number of factors coming into play in a phenomenological complex is too large, scientific method in most cases fails us. One need only think of the weather, in which case prediction even for a few days ahead is impossible. Nevertheless no one doubts that we are confronted with a causal connection whose causal components are in the main known to us.

Occurrences in this domain are beyond the reach of exact prediction because of the variety of factors in operation, not because of any lack of order in nature." (Albert Einstein, "Science and Religion", 1941)

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

"A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended its area of applicability." (Albert Einstein, "Autobiographical Notes", 1949)

"In speaking here of ‘comprehensibility’, the expression is used in its most modest sense. It implies: the production being produced by the creation of general concepts, relations between these concepts and sense experience. It is in this sense that the world of our sense experiences is comprehensible. The fact that it is comprehensible is a miracle." (Albert Einstein, "Out of My Later Years", 1950)

"Physics too deals with mathematical concepts; however, these concepts attain physical content only by the clear determination of their relation to the objects of experience." (Albert Einstein, "Out of My Later Years", 1950)

"Science is the attempt to make the chaotic diversity of our sense-experience correspond to a logically uniform system of thought."  (Albert Einstein, "Out of My Later Years", 1950)

"Space-time does not claim existence on its own, but only as a structural quality of the field." (Albert Einstein, 1954)

"The important point for us to observe is that all these constructions and the laws connecting them can be arrived at by the principle of looking for the mathematically simplest concepts and the link between them. In the limited number of mathematically existent simple field types, and the simple equations possible between them, lies the theorist’s hope of grasping the real in all its depth." (Albert Einstein, "Ideas and Opinions", 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) 

"The theory of relativity is a fine example of the fundamental character of the modern development of theoretical science. The initial hypotheses become steadily more abstract and remote from experience. On the other hand, it gets nearer to the grand aim of all science, which is to cover the greatest possible number of empirical facts by logical deduction from the smallest possible number of hypotheses or axioms." (Albert Einstein, 1954)

"We have thus assigned to pure reason and experience their places in a theoretical system of physics. The structure of the system is the work of reason: the empirical contents and their mutual relations must find their representation in the conclusions of the theory. In the possibility of such a representation lie the sole value and justification of the whole system, and especially of the concepts and fundamental principles which underlie it. Apart from that, these latter are free inventions of human intellect, which cannot be justified either by the nature of that intellect or in any other fashion a priori." (Albert Einstein, "Ideas and Opinions", 1954)

"What distinguishes the language of science from language as we ordinarily understand the word? […] What science strives for is an utmost acuteness and clarity of concepts as regards their mutual relation and their correspondence to sensory data." (Albert Einstein, "Ideas and Opinions", 1954)

"When a man after long years of searching chances upon a thought which discloses something of the beauty of this mysterious universe, he should not therefore be personally celebrated. He is already sufficiently paid by his experience of seeking and finding." (Albert Einstein, [The New York Times] 1978)

 "All great achievements in science start from intuitive knowledge, namely, in axioms, from which deductions are then made. […] Intuition is the necessary condition for the discovery of such axioms. " (Albert Einstein) 

 "Equations are more important to me, because politics is for the present, but an equation is something for eternity. " (Albert Einstein)

"How can it be that mathematics, a product of human thought independent of experience, is so admirably adapted to the objects of reality." (Albert Einstein)

 "It is a miracle that curiosity survives formal education. " (Albert Einstein)

 "It is an outcome of faith that nature - as she is perceptible to our five senses - takes the character of such a well formulated puzzle. " (Albert Einstein)

 "It stands to the everlasting credit of science that by acting on the human mind it has overcome man's insecurity before himself and before nature. " (Albert Einstein)

"It’s not that I’m so smart, it’s just that I stay with problems longer." (Albert Einstein)

"Look deep, deep, deep into nature, and then you will understand everything." (Albert Einstein)

"Our experience hitherto justifies us in believing that nature is the realization of the simplest conceivable mathematical ideas. " (Albert Einstein)

 "That deep emotional conviction of the presence of a superior reasoning power, which is revealed in the incomprehensible universe, forms my idea of God. " (Albert Einstein)

"The aim [of education] must be the training of independently acting and thinking individuals who, however, see in the service to the community their highest life achievement." (Albert Einstein)

"The supreme task 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 lead to them." (Albert Einstein)

"The truth of a theory is in your mind, not in your eyes." (Albert Einstein)

 "Truth is what stands the test of experience. " (Albert Einstein)

"To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advance in science." (Albert Einstein)

 "We can not solve our problems with the same level of thinking that created them." (Albert Einstein)

 "What I’m really interested in is whether God could have made the world in a different way; that is, whether the necessity of logical simplicity leaves any freedom at all. " (Albert Einstein)

Karl R Popper - Collected Quotes

"A scientist, whether theorist or experimenter, puts forward statements, or systems of statements, and tests them step by step. In the field of the empirical sciences, more particularly, he constructs hypotheses, or systems of theories, and tests them against experience by observation and experiment." (Karl R Popper, "The Logic of Scientific Discovery", 1934)

“It is sometimes said of two expositions of one and the same mathematical proof that the one is simpler or more elegant than the other. This is a distinction which has little interest from the point of view of the theory of knowledge; it does not fall within the province of logic, but merely indicates a preference of an aesthetic or pragmatic character.” (Karl R Popper, “The Logic of Scientific Discovery”, 1934)

"Modern positivists are apt to see more clearly that science is not a system of concepts but rather a system of statements." (Karl R Popper, "The Logic of Scientific Discovery", 1934)

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

"Science does not aim, primarily, at high probabilities. It aims at a high informative content, well backed by experience. But a hypothesis may be very probable simply because it tells us nothing, or very little." (Karl R Popper, "The Logic of Scientific Discovery", 1934)

"The most important application of the theory of probability is to what we may call 'chance-like' or 'random' events, or occurrences. These seem to be characterized by a peculiar kind of incalculability which makes one disposed to believe - after many unsuccessful attempts - that all known rational methods of prediction must fail in their case. We have, as it were, the feeling that not a scientist but only a prophet could predict them. And yet, it is just this incalculability that makes us conclude that the calculus of probability can be applied to these events." (Karl R Popper, "The Logic of Scientific Discovery", 1934)

"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 R Popper, "The logic of scientific discovery", 1934)

"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)  

“It is easy to obtain confirmations, or verifications, for nearly every theory - if we look for confirmations. Confirmations should count only if they are the result of risky predictions. […] A theory which is not refutable by any conceivable event is non-scientific. Irrefutability is not a virtue of a theory (as people often think) but a vice. Every genuine test of a theory is an attempt to falsify it, or refute it.” (Karl R Popper, “Conjectures and Refutations: The Growth of Scientific Knowledge”, 1963)

"All prescientific knowledge, whether animal or human, is dogmatic; and science begins with the invention of the non-dogmatic, critical method." (Karl R Popper, "The Logic and Evolution of Scientific Theory", [in "All Life is Problem Solving", 1999] 1972)

"An empirical scientific theory differs from other theories because it may be undone by possible experimental results: that is to say, possible experimental results can be described that would falsify the theory if we were actually to obtain them." (Karl R Popper, "The Logic and Evolution of Scientific Theory", [in "All Life is Problem Solving", 1999] 1972)

"At any event, the critical approach is the crucial novelty that makes science what it is, achieved above all through objective, public, linguistic formulation of its theories. This usually leads to a taking of sides and hence to critical discussion." (Karl R Popper, "The Logic and Evolution of Scientific Theory", [in "All Life is Problem Solving", 1999] 1972)

"But the laws of addition and multiplication (the associative laws, for example) are not a human invention. They are unintended consequences of human invention, and they were discovered. And the existence ofprime numbers - indivisible numbers that are the product only of themselves and unity - is also a discovery, no doubt quite a late one. The prime numbers were discovered in the series of natural numbers, not by everyone but by people who studied these numbers and their special peculiarities - by real mathematicians." (Karl R Popper, "Notes of a Realist on the Body-Mind Problem", [in "All Life is Problem Solving", 1999] 1972)

"Higher organisms are able to learn through trial and error how a certain problem should be solved. We may say that they too make testing movements - mental testings - and that to learn is essentially to tryout one testing movement after another until one is found that solves the problem. We might compare the animal's successful solution to an expectation and hence to a hypothesis or a theory. For the animal's behaviour shows us that it expects (perhaps unconsciously or dispositionally) that in a similar case the same testing movements will again solve the problem in question." (Karl R Popper, "The Logic and Evolution of Scientific Theory", [in "All Life is Problem Solving", 1999] 1972)

"Science is a product of the human mind, but this product is as objective as a cathedral." (Karl R Popper, "The Logic and Evolution of Scientific Theory", [in "All Life is Problem Solving", 1999] 1972)

"The idea of approximation to the truth is, in my view, one of the most important ideas in the theory ofscience. [...] The idea of approximation to the truth - like the idea of truth as a

regulative principle - presupposes a realistic view ofthe world. It does not presuppose that reality is as our scientific theories describe it; but it does presuppose that there is a reality and that we and our theories - which are ideas we have ourselves created and are therefore always idealizations - can draw closer and closer to an adequate description of reality, if we employ the four-stage method of trial and error." (Karl R Popper, "The Logic and Evolution of Scientific Theory", [in "All Life is Problem Solving", 1999] 1972)

"The natural as well as the social sciences always start from problems, from the fact that something inspires amazement in us, as the Greek philosophers used to say. To solve these problems, the sciences use fundamentally the same method that common sense employs, the method of trial and error. To be more precise, it is the method of trying out solutions to our problem and then discarding the false ones as erroneous. This method assumes that we work with a large number of experimental solutions. One solution after another is put to the test and eliminated." (Karl R Popper, "The Logic and Evolution of Scientific Theory", [in "All Life is Problem Solving", 1999] 1972)

"The realistic view ofthe world, together with the idea of approximation to the truth, seem to me indispensable for an understanding of the perpetually idealizing character of science." (Karl R Popper, "The Logic and Evolution of Scientific Theory", [in "All Life is Problem Solving", 1999] 1972)

"Science begins with problems. It attempts to solve them through bold, inventive theories. The great majority of theories are false and/or untestable. Valuable, testable theories will search for errors. We try to find errors and to eliminate them. This is science: it consists of wild, often irresponsible ideas that it places under the strict control of error correction."(Karl R Popper, "Epistemology and the Problem of Peace", [lecture in "All Life is Problem Solving", 1999] 1985)

"Scientists, like all organisms, work with the method of trial and error. The trial is a solution to a problem. In the evolution of the plant or animal kingdom, error or, to be more precise, the correction of error usually means eradication of the organism; in science it usually means eradication of the hypothesis or theory." (Karl R Popper, "Epistemology and the Problem of Peace", [lecture in "All Life is Problem Solving", 1999] 1985)

"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)

"The method of natural science is the conscious search for errors and correction of them through conscious criticism. Ideally such criticism should be impersonal and directed only at the theories or hypotheses in question."  (Karl R Popper, "Epistemology and the Problem of Peace", [lecture in "All Life is Problem Solving", 1999] 1985)

"We can assert the truth, attain the truth, often enough. But we can never attain certainty." (Karl R Popper, "Epistemology and the Problem of Peace", [lecture in "All Life is Problem Solving", 1999] 1985)

"[...] everything we know is genetically a priori. All that is a posteriori is the selection from what we ourselves have invented a priori." (Karl R Popper, "The Epistemological Position of Evolutionary Epistemology", [in "All Life is Problem Solving", 1999] 1987)

"The task of us all as thinking human beings is to discover the truth. The truth is absolute and objective, but we do not have it in the bag. We are constantly seeking it and often find it only with difficulty; and we keep trying to improve our approximation to the truth. If truth were not absolute and objective, we should not be able to go wrong. Or our mistakes would be as good as our truth." (Karl R Popper, "The Epistemological Position of Evolutionary Epistemology", [in "All Life is Problem Solving", 1999] 1987)

"We know nothing - that is the first point. Therefore we should be very modest - that is the second. That we should not claim to know when we do not know - that is

the third." (Karl R Popper, "The Epistemological Position of Evolutionary Epistemology", [in "All Life is Problem Solving", 1999] 1987)

"[...] we learn only through trial and error. Our trials, however, are always our hypotheses. They stem from us, not from the external world. All we learn from the external world is that some of our efforts are mistaken." (Karl R Popper, "The Epistemological Position of Evolutionary Epistemology", [in "All Life is Problem Solving", 1999] 1987)

"Without intuition we get nowhere - even though most of our intuitions eventually turn out wrong. We need intuitions, ideas, if possible, competing ideas; and we need ideas about how those ideas can be criticized, improved, and critically tested. And until they are refuted (indeed, for longer), we must also put up with questionable ideas. For even the best ideas are questionable." (Karl R Popper, "Kepler's Metaphysics of the Solar System and His Empirical Criticism", [in "All Life is Problem Solving", 1999] 1986)

"Classical models tell us more than we at first can know." (Karl R Popper)

"A theory is just a mathematical model to describe the observations." (Karl R Popper)

"Science starts from problems, and not from observations." (Karl R Popper)

David Hilbert - Collected Quotes

"A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street." (David Hilbert [paraphrasing Joseph D Gergonne], "Mathematical Problems", 1900)

"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)

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

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

"Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock our efforts. It should be to us a guidepost on the mazy path to hidden truths, and ultimately a reminder of our pleasure in the successful solution." (David Hilbert, "Mathematical Problems", [lecture at Paris International Congress] 1900)

"Occasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense, and for this reason do not succeed. The problem then arises: to show the impossibility of the solution   under the given hypotheses, or in the sense contemplated. Such proofs of impossibility were effected by the ancients, for instance when they showed that the ratio of the hypotenuse to the side of an isosceles triangle is irrational. In later mathematics, the question as to the impossibility of certain solutions plays a preeminent pan, and we perceive in this way that old and difficult problems, such as the proof of the axiom of parallels, the squaring of the circle, or the solution of equations of the fifth degree by radicals have finally found fully satisfactory and rigorous solutions, although in another sense than that originally intended. It is probably this important fact along with other philosophical reasons that gives rise to conviction (which every mathematician shares, but which no one has as yet supported by a proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts." (David Hilbert, "Mathematical Problems", [lecture at Paris International Congress] 1900) 

“Besides it is an error to believe that rigour is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof.” (David Hilbert, “Mathematical Problems”, Bulletin of the American Mathematical Society Vol. 8, 1902)

"He who seeks for methods without having a definite problem in mind seeks for the most part in vain." (David Hilbert, 1902)

“With the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and to cast aside some more complicated developments. It is therefore possible for the individual investigator, when he makes these sharper tools and simpler methods his own, to find his way more easily in the various branches of mathematics than is possible in any other science.” (David Hilbert, “Mathematical Problems”, Bulletin of the American Mathematical Society Vol. 8, 1902)

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

“To reach our goal [of proving consistency], we must make the proofs as such the object of our investigation; we are thus compelled to a sort of proof theory which studies operations with the proofs themselves.” (David Hilbert, 1922)

"The axioms and provable theorems (i.e. the formulas that arise in this alternating game [namely formal deduction and the adjunction of new axioms]) are images of the thoughts that make up the usual procedure of traditional mathematics; but they are not themselves the truths in the absolute sense. Rather, the absolute truths are the insights (Einsichten) that my proof theory furnishes into the provability and the consistency of these formal systems." (David Hilbert; “Die logischen Grundlagen der Mathematik.“ Mathematische Annalen 88 (1), 1923)

“For it is true, generally speaking, that mathematics is not a popular subject, even though its importance may be generally conceded. The reason for this is to be found in the common superstition that mathematics is but a continuation, a further development, of the fine art of arithmetic, of juggling with numbers.” (David Hilbert, “Anschauliche Geometrie”, 1932)

“In mathematics […] we find two tendencies present. On the one hand, the tendency towards abstraction seeks to crystallise the logical relations inherent in the maze of materials [….] being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency towards intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations.” (David Hilbert, “Geometry and the imagination”, 1952)

“Mathematics is not a popular subject, even though its importance may be generally conceded. The reason for this is to be found in the common superstition that mathematics is but a continuation, a further development, of the fine art of arithmetic, of juggling with numbers.” (David Hilbert, “Geometry and the Imagination”, 1952)

“In topology we are concerned with geometrical facts that do not even involve the concepts of a straight line or plane but only the continuous connectiveness between points of a figure.” (David Hilbert, “Geometry and Imagination”, 1952)

“The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.” (David Hilbert)

“When the answer to a mathematical problem cannot be found, then the reason is frequently that we have not recognized the general idea from which the given problem only appears as a link in a chain of related problems.” (David Hilbert)

“This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by reason, for in mathematics there is no ignorabimus." (David Hilbert)

“The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena." (David Hilbert) 
"For, compared with the immense expanse of modern mathematics, what would the wretched remnants mean, the few isolated results incomplete and unrelated, that the intuitionists have obtained." (David Hilbert)

"As long as a branch of science offers an abundance of problems, so long is it alive." (David Hilbert)

"There is nothing more practical than a good theory.” ([attributed to] David Hilbert)

Richard P Feynman - Collected Quotes

"Each piece, or part, of the whole of nature is always merely an approximation to the complete truth, or the complete truth so far as we know it. In fact, everything we know is only some kind of approximation, because we know that we do not know all the laws as yet. Therefore, things must be learned only to be unlearned again or, more likely, to be corrected." (Richard Feynman, "The Feynman Lectures on Physics" Vol. 1,1964)

"The 'underlying unity' might mean that everything is made out of the same stuff, and therefore obeys the same equations. That sounds like a good explanation, but let us think. The electrostatic potential, the diffusion of neutrons, heat flow - are we really dealing with the same stuff? Can we really imagine that the electrostatic potential is physically identical to the temperature, or to the density of particles? [...] The displacement of a membrane is certainly not like a temperature. Why, then, is there 'an underlying unity'? [...] Is it possible that this is the clue? That the thing which is common to all the phenomena is the space, the framework into which the physics is put? As long as things are reasonably smooth in space, then the important things that will be involved will be the rates of change of quantities with position in space. That is why we always get an equation with a gradient. [...] What is common to all our problems is that they involve space. "  (Richard Feynman, "Lecture Notes on Physics" Vol. 3, 1964)

"Why are the equations from different phenomena so similar? We might say: ‘It is the underlying unity of nature.’ But what does that mean? What could such a statement mean? It could mean simply that the equations are similar for different phenomena; but then, of course, we have given no explanation. The underlying unity might mean that everything is made out of the same stuff, and therefore obeys the same equations." (Richard P Feynman, "Lecture Notes on Physics" Vol. III, 1964)

"In its efforts to learn as much as possible about nature, modem physics has found that certain things can never be ‘known’ with certainty. Much of our knowledge must always remain uncertain. The most we can know is in terms of probabilities." (Richard P Feynman, "The Feynman Lectures on Physics", 1964)

"Another thing I must point out is that you cannot prove a vague theory wrong. If the guess that you make is poorly expressed and rather vague, and the method that you use for figuring out the consequences is a little vague - you are not sure, and you say, 'I think everything's right because it's all due to so and so, and such and such do this and that more or less, and I can sort of explain how this works' […] then you see that this theory is good, because it cannot be proved wrong! Also if the process of computing the consequences is indefinite, then with a little skill any experimental results can be made to look like the expected consequences." (Richard P Feynman, "The Character of Physical Law", 1965)

"If science is to progress, what we need is the ability to experiment, honestly in reporting the results - the results must be reported without somebody saying what they would like the results to have been - and finally - an important thing - the intelligence to interpret the results. An important point about this intelligence is that it should not be sure ahead of time what must be. It cannot be prejudiced, and say 'That is very unlikely; I don’t like that.'" (Richard P Feynman, "The Character of Physical Law", 1965)

"So in order to understand the physics one must always have a neat balance and contain in his head all of the various propositions and their interrelationships because the laws often extend beyond the range of their deductions. This will only have no importance when all the laws are known." (Richard Feynman, "The Character of Physical Law", 1965)

"This is the key of modern science and it was the beginning of the true understanding of Nature - this idea to look at the thing, to record the details, and to hope that in the information thus obtained might lie a clue to one or another theoretical interpretation." (Richard P Feynman, "The Character of Physical Law", 1965)

 "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)

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

"The method of guessing the equation seems to be a pretty effective way of guessing new laws. This shows again that mathematics is a deep way of expressing nature, and any attempt to express nature in philosophical principles, or in seat-of-the-pants mechanical feelings, is not an efficient way." (Richard Feynman, "The Character of Physical Law", 1965)

"To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature. […] If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in." (Richard P Feynman, "The Character of Physical Law", 1965)

"We decided that ‘trivial’ means ‘proved’. So, we joked with the mathematicians: "We have a new theorem - that mathematicians can prove only trivial theorems, because every theorem that’s proved is trivial." (Richard P Feynman, "Surely You’re Joking, Mr. Feynman!: Adventures of a Curious Character", 1985)

"[…] the more you see how strangely Nature behaves, the harder it is to make a model that explains how even the simplest phenomena actually work. So theoretical physics has given up on that." (Richard P Feynman, "QED: The Strange Theory of Light and Matter", 1985)

"It is our responsibility as scientists, knowing the great progress which comes from a satisfactory philosophy of ignorance, the great progress which is the fruit of freedom of thought, to proclaim the value of this freedom; to teach how doubt is not to be feared but welcomed and discussed; and to demand this freedom as our duty to all coming generations." (Richard P (Feynman, "What Do You Care What Other People Think?", 1988)

"The principle of science, the definition, almost, is the following: The test of all knowledge is experiment. Experiment is the sole judge of scientific ‘truth’." (Richard Feynman, "Six Easy Pieces", 1994)

"The rate of the development of science is not the rate at which you make observations alone but, much more important, the rate at which you create new things to test." (Richard Feynman, "The Meaning of It All", 1998)

"When the scientist tells you he does not know the answer, he is an ignorant man. When he tells you he has a hunch about how it is going to work, he is uncertain about it. When he is pretty sure of how it is going to work, and he tells you, 'This is the way it is going to work, I’ll bet', he still is in some doubt. And it is of paramount importance, in order to make progress, that we recognize this ignorance and this doubt. Because we have the doubt, we then propose looking in new directions for new ideas. The rate of development in science is not the rate at which you make observations alone but, much more important, the rate at which you create new things to test." (Richard P Feynman, "The Meaning of It All: Thoughts of a Citizen Scientist", 1998)

"Mathematics is trivial, but I can’t do my work without it." (Richard Feynman)

"Science is a way of trying not to fool yourself. The first principle is that you must not fool yourself, and you are the easiest person to fool." (Richard Feynman) [attributed to]

"One measure of our understanding is the number of independent ways we are able to get to the same result." (Richard P Feynman)

On Theories (1860-1874)

“The world little knows how many of the thoughts and theories which have passed through the mind of a scientific investigator have been crushed in silence and secrecy; that in the most successful instances not a tenth of the suggestions, the hopes, the wishes, the preliminary conclusions have been realized.” (Michael Faraday, “The Forces of Matter”, 1860)

“Observation is so wide awake, and facts are being so rapidly added to the sum of human experience, that it appears as if the theorizer would always be in arrears, and were doomed forever to arrive at imperfect conclusion; but the power to perceive a law is equally rare in all ages of the world, and depends but little on the number of facts observed.” (Henry Thoreau, “A Week on the Concord and Merrimack Rivers”, 1862) 

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

"Science asks no questions about the ontological pedigree or a priori character of a theory, but is content to judge it by its performance; and it is thus that a knowledge of nature, having all the certainty which the senses are competent to inspire, has been attained - a knowledge which maintains a strict neutrality toward all philosophical systems and concerns itself not with the genesis or a priori grounds of ideas." (Chauncey Wright, "The Philosophy of Herbert Spencer", North American Review, 1865)

 “Step by step we cross great eras in the development of thought: there is no sudden gigantic stride; a theory proceeds by slow evolution until it dominates or is destroyed.” (George F Rodwell, “Theory of Phlogiston”, ‘The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science’ 35, 1868)

"Isolated facts and experiments have in themselves no value, however great their number may be. They only become valuable in a theoretical or practical point of view when they make us acquainted with the law of a series of uniformly recurring phenomena, or, it may be, only give a negative result showing an incompleteness in our knowledge of such a law, till then held to be perfect." (Hermann von Helmholtz, "The Aim and Progress of Physical Science", 1869)

 “The triumph of a theory is to embrace the greatest number and the greatest variety of facts.” (Charles A Wurtz, “A History of Chemical Theory from the Age of Lavoisier to the Present Time”, 1869)

“The aim of natural science is to obtain connections among phenomena. Theories, however, are like withered leaves, which drop off after having enabled the organism of science to breathe for a time." (Ernst Mach, “Die Geschichte und die Wurzel des Satzes von der Erhaltung der Arbeit”, 1871)

On Theories (1875-1899)

“The possession of an original theory which has not yet been assailed must certainly sweeten the temper of a man who is not beforehand ill-natured.” (George Eliot, “Theophrastus Such”, 1879)

"Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hypothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims; and neither law can rule nor theory explain without the sanction of mathematics.” (Benjamin Peirce, “Linear Associative Algebra”, American Journal of Mathematics, Vol. 4, 1881)

"As for everything else, so for a mathematical theory: beauty can be perceived but not explained." (Arthur Cayley, [president's address] 1883)

"It would be an error to suppose that the great discoverer seizes at once upon the truth, or has any unerring method of divining it. In all probability the errors of the great mind exceed in number those of the less vigorous one. Fertility of imagination and abundance of guesses at truth are among the first requisites of discovery; but the erroneous guesses must be many times as numerous as those that prove well founded. The weakest analogies, the most whimsical notions, the most apparently absurd theories, may pass through the teeming brain, and no record remain of more than the hundredth part. […] The truest theories involve suppositions which are inconceivable, and no limit can really be placed to the freedom of hypotheses." (W Stanley Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1877)

“Perfect readiness to reject a theory inconsistent with fact is a primary requisite of the philosophic mind. But it, would be a mistake to suppose that this candour has anything akin to fickleness; on the contrary, readiness to reject a false theory may be combined with a peculiar pertinacity and courage in maintaining an hypothesis as long as its falsity is not actually apparent.” (William S Jevons, “The Principles of Science”, 1887)

“The history of thought should warn us against concluding that because the scientific theory of the world is the best that has yet been formulated, it is necessarily complete and final. We must remember that at bottom the generalizations of science or, in common parlance, the laws of nature are merely hypotheses devised to explain that ever-shifting phantasmagoria of thought which we dignify with the high-sounding names of the world and the universe.” (Sir James G Frazer, “The Golden Bough: A Study in Magic and Religion”, 1890)

"The study of theory must go hand in hand with that of facts: and for dealing with most modern problems it is modern facts that are of the greatest use." (Alfred Marshall, "Principles of Economics", 1890)

“One is almost tempted to assert that quite apart from its intellectual mission, theory is the most practical thing conceivable, the quintessence of practice as it were, since the precision of its conclusions cannot be reached by any routine of estimating or trial and error; although given the hidden ways of theory, this will hold only for those who walk them with complete confidence.” (Ludwig E Boltzmann, “On the Significance of Theories”, 1890) 

“Facts are not much use, considered as facts. They bewilder by their number and their apparent incoherency. Let them be digested into theory, however, and brought into mutual harmony, and it is another matter. Theory is of the essence of facts. Without theory scientific knowledge would be only worthy of the mad house.” (Oliver Heaviside, “Electromagnetic Theory”, 1893)

”Scientific facts accumulate rapidly, and give rise to theories with almost equal rapidity. These theories are often wonderfully enticing, and one is apt to pass from one to another, from theory to theory, without taking care to establish each before passing on to the next, without assuring oneself that the foundation on which one is building is secure. Then comes the crash; the last theory breaks down utterly, and on attempting to retrace our steps to firm ground and start anew, we may find too late that one of the cards, possibly at the very foundation of the pagoda, is either faultily placed or in itself defective, and that this blemish easily remedied if detected in time has, neglected, caused the collapse of the whole structure on whose erection so much skill and perseverance have been spent.” (Arthur M Marshall, 1894)

On Theories (1940-1949)

"There is nothing as practical as a good theory” (Kurt Z Lewin, "Psychology and the process of group living", Journal of Social Psychology 17, 1943)

“To a scientist a theory is something to be tested. He seeks not to defend his beliefs, but to improve them. He is, above everything else, an expert at ‘changing his mind’.” (Wendell Johnson, 1946)

"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)

“One expects a mathematical theorem or a mathematical theory not only to describe and to classify in a simple and elegant way numerous and a priori disparate special cases. One also expects ‘elegance’ in its ‘architectural’ structural makeup.” (John von Neumann, "The Mathematician" [in "Works of the Mind" Vol. I (1), 1947]) 

”We can put it down as one of the principles learned from the history of science that a theory is only overthrown by a better theory, never merely by contradictory facts.” (James B Conant, “On Understanding Science”, 1947)

"A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended its area of applicability." (Albert Einstein, "Autobiographical Notes", 1949)

“When a scientific theory is firmly established and confirmed, it changes its character and becomes a part of the metaphysical background of the age: a doctrine is transformed into a dogma.” (Max Born, “Natural Philosophy of Cause and Chance”, 1949)

On Intuition (1970-1979)

“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)

“The mathematically formulated laws of quantum theory show clearly that our ordinary intuitive concepts cannot be unambiguously applied to the smallest particles. All the words or concepts we use to describe ordinary physical objects, such as position, velocity, color, size, and so on, become indefinite and problematic if we try to use them of elementary particles.” (Werner K Heisenberg, “Across the Frontiers”, 1974)

“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)

“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)