Reuben Hersh - Collected Quotes

"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)

"A mathematical entity is a concept, a shared thought. Once you have acquired it, you have it available, for inspection or manipulation. If you understand it correctly (as a student, or as a professional) your ‘mental model’ of that entity, your personal representative of it, matches those of others who understand it correctly. (As is verified by giving the same answers to test questions.) The concept, the cultural entity, is nothing other than the collection of the mutually congruent personal representatives, the ‘mental models’, possessed by those participating in the mathematical culture." (Reuben Hersh, "Experiencing Mathematics: What Do We Do, when We Do Mathematics?", 2014)

"A coherent inclusive study of the nature of mathematics would contribute to our understanding of problem-solving in general. Solving problems is how progress is made in all of science and technology. The synthesizing energy to achieve such a result would be a worthy and inspiring task for philosophy." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)

"A mathematician possesses a mental model of the mathematical entity she works on. This internal mental model is accessible to her direct observation and manipulation. At the same time, it is socially and culturally controlled, to conform to the mathematics community's collective model of the entity in question. The mathematician observes a property of her own internal model of that mathematical entity. Then she must find a recipe, a set of instructions, that enables other competent, qualified mathematicians to observe the corresponding property of their corresponding mental model. That recipe is the proof. It establishes that property of the mathematical entity." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)

"Different models are both competitive and complementary. Their standing will depend on their benefits in practice. If philosophy of mathematics were seen as modeling rather than as taking positions, it might consider paying attention to mathematics research and mathematics teaching as testing grounds for its models." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)

"History of mathematics is done by mathematicians as well as historians. History models mathematics as a segment of the ongoing story of human culture. Mathematicians are likely to see the past through the eyes of the present, and ask, ‘Was it important? natural? deep? surprising? elegant?’ The historian sees mathematics as a thread in the ever-growing web of human life, intimately interwoven with finance and technology, with war and peace. Today's mathematics is the culmination of all that has happened before now, yet to future viewpoints it will seem like a brief, outmoded stage of the past." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)

"Logic sees mathematics as a collection of virtual inscriptions - declarative sentences that could in principle be written down. On the basis of that vision, it offers a model: formal deductions from formal axioms to formal conclusions--formalized mathematics. This vision itself is mathematical. Mathematical logic is a branch of mathematics, and whatever it's saying about mathematics, it is saying about itself--self-reference." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)

"[...] mathematics is too complex, varied and elaborate to be encompassed in any model. An all-inclusive model would be like the map in the famous story by Borges - perfect and inclusive because it was identical to the territory it was mapping." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017) 

"Mathematical modeling is the modern version of both applied mathematics and theoretical physics. In earlier times, one proposed not a model but a theory. By talking today of a model rather than a theory, one acknowledges that the way one studies the phenomenon is not unique; it could also be studied other ways. One's model need not claim to be unique or final. It merits consideration if it provides an insight that isn't better provided by some other model." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)

Democritus - Colltected Quotes

"By convention sweet, by convention bitter; by convention hot, by convention cold, by convention color: but in reality, atoms and void." (Democritus)

"By desiring little, a poor man makes himself rich." (Democritus)

"Everything existing in the universe is the fruit of chance and necessity." (Democritus)

"Hope of ill gain is the beginning of loss." (Democritus)

"I would rather discover one true cause than gain the kingdom of Persia." (Democritus)

"If thou suffer injustice, console thyself; the true unhappiness is in doing it." (Democritus)

"It is better to destroy one's own errors than those of others." (Democritus)

"It is godlike ever to think on something beautiful and on something new." (Democritus)

"It is greed to do all the talking but not to want to listen at all." (Democritus)

"Men should strive to think much and know little." (Democritus)

"Nothing exists except atoms and empty space; everything else is opinion." (Democritus)

"Neither art nor wisdom may be attained without learning."  (Democritus)

"No power and no treasure can outweigh the extension of our knowledge." (Democritus)

"Nothing is so easy as to deceive one's self; for what we wish, that we readily believe." (Democritus)

"Raising children is an uncertain thing; success is reached only after a life of battle and worry." (Democritus)

"The first principles of the universe are atoms and empty space; everything else is merely thought to exist." (Democritus)

"The wrongdoer is more unfortunate than the man wronged." (Democritus)

"There are many who know many things, yet are lacking in wisdom." (Democritus)

"Throw moderation to the winds, and the greatest pleasures bring the greatest pains." (Democritus)

Max Born - Collected Quotes

"The difficulty involved in the proper and adequate means of describing changes in continuous deformable bodies is the method of differential equations. […] They express mathematically the physical concept of contiguous action." (Max Born, "Einstein's Theory of Relativity", 1920)

"It is natural that a man should consider the work of his hands or his brain to be useful and important. Therefore nobody will object to an ardent experimentalist boasting of his measurements and rather looking down on the 'paper and ink' physics of his theoretical friend, who on his part is proud of his lofty ideas and despises the dirty fingers of the other." (Max Born, " Experiment and Theory in Physics", 1943)

"The conception of chance enters in the very first steps of scientific activity in virtue of the fact that no observation is absolutely correct. I think chance is a more fundamental conception that causality; for whether in a concrete case, a cause-effect relation holds or not can only be judged by applying the laws of chance to the observation." (Max Born, 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)

"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)

"Every object that we perceive appears in innumerable aspects. The concept of the object is the invariant of all these aspects. From this point of view, the present universally used system of concepts in which particles and waves appear simultaneously, can be completely justified. The latest research on nuclei and elementary particles has led us, however, to limits beyond which this system of concepts itself does not appear to suffice. The lesson to be learned from what I have told of the origin of quantum mechanics is that probable refinements of mathematical methods will not suffice to produce a satisfactory theory, but that somewhere in our doctrine is hidden a concept, unjustified by experience, which we must eliminate to open up the road." (Max Born, "The Statistical Interpretations of Quantum Mechanics", [Nobel lecture] 1954)

"[...] if we can never actually determine more than one of the two properties (possession of a definite position and of a definite momentum), and if when one is determined we-can make no assertion at all about the other property for the same moment, so far as our experiment goes, then we are not justified in concluding that the 'thing' under examination can actually be described as a particle in the usual sense of the term." (Max Born, "Atomic Physics", 1957)

"Physics is in the nature of the case indeterminate, and therefore the affair of statistics." (Max Born, "Atomic Physics", 1957)

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

"[...] the whole course of events is determined by the laws of probability; to a state in space there corresponds a definite probability, which is given by the de Brogile wave associated with the state." (Max Born, "Atomic Physics", 1957)

"The belief that there is only one truth and that oneself is in possession of it, seems to me the deepest root of all that is evil in the world." (Max Born, "Natural Philosophy of Cause and Chance", 1964)

"There are metaphysical problems, which cannot be disposed of by declaring them meaningless. For, as I have repeatedly said, they are ‘beyond physics’ indeed and demand an act of faith. We have to accept this fact to be honest. There are two objectionable types of believers: those who believe the incredible and those who believe that ‘belief’ must be discarded and replaced by "the scientific method." (Max Born, "Natural Philosophy of Cause and Chance", 1964)

"Science is not formal logic - it needs the free play of the mind in as great a degree as any other creative art. It is true that this is a gift which can hardly be taught, but its growth can be encouraged in those who already possess it." (Max Born)

Georg Cantor - Collected Quotes

"If two well-defined manifolds M and N can be coordinated with each other univocally and completely, element by element (which, if possible in one way, can always happen in many others), we shall employ in the sequel the expression, that those manifolds have the same power or, also, that they are equivalent." (Georg Cantor, "Ein Beitrag zur Mannigfaltigkeitslehre", 1878)

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

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

"By a manifold or a set I understand in general every Many that can be thought of as One, i.e., every collection of determinate elements which can be bound up into a whole through a law, and with this I believe to define something that is akin to the Platonic εἷδος [form] or ἷδεα [idea]." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883) 

"If we now notice that all of the numbers previously obtained and their next successors fulfill a certain condition, [that the set of their predecessors is denumerable,] then this condition offers itself, if it is imposed as a requirement on all numbers to be formed next, as a new third principle [...] which I shall call principle of restriction or limitation and which, as I shall show, yields the result that the second number-class (II) defined with its assistance not only has a higher power than [the first number-class] (I), but precisely the next higher, that is, the second power." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883) 

"In order for there to be a variable quantity in some mathematical study, the domain of its variability must strictly speaking be known beforehand through a definition. However, this domain cannot itself be something variable, since otherwise each fixed support for the study would collapse. Thus this domain is a definite, actually infinite set of values. Hence each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite." (Georg Cantor, "Über die verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen" ["Over the different views with regard to the actual infinite numbers"], 1886)

"There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual infinite can be demonstrated." (Georg Cantor, "Über die verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen" ["Over the different views with regard to the actual infinite numbers"], 1886) 

"A set is a Many that allows itself to be thought of as a One." (Georg Cantor)

"An infinite set is one that can be put into a one-to-one correspondence with a proper subset of itself." (Georg Cantor)

"Every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph as its cardinal number." (Georg Cantor)

"I realise that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers." (Georg Cantor)

"In particular, in introducing new numbers, mathematics is only obliged to give definitions of them, by which such a definiteness and, circumstances permitting, such a relation to the older numbers are conferred upon them that in given cases they can definitely be distinguished from one another. As soon as a number satisfies all these conditions, it can and must be regarded as existent and real in mathematics. Here I perceive the reason why one has to regard the rational, irrational, and complex numbers as being just as thoroughly existent as the finite positive integers." (Georg Cantor)

"My theory stands as firm as a rock; every arrow directed against it will quickly return to the archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things." (Georg Cantor)

"To ask the right question is harder than to answer it." (Georg Cantor)

"The essence of mathematics lies in its freedom." (Georg Cantor)

"The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds." (Georg Cantor)

Niels Bohr - Collected Quotes

"We must be clear that when it comes to atoms, language can be used only as in poetry. The poet, too, is not nearly so concerned with describing facts as with creating images and establishing mental connections." (Niels Bohr, 1920) 

"[...] an independent reality in the ordinary' physical sense can be ascribed neither to the phenomena nor to the agencies of observation." (Niels Bohr, "Atomic Theory and the Description of Nature", 1934)

"The great extension of our experience in recent years has brought light to the insufficiency of our simple mechanical conceptions and, as a consequence, has shaken the foundation on which the customary interpretation of observation was based." (Niels Bohr, "Atomic Physics and the Description of Nature", 1934)

"[...] in quantum mechanics, we are not dealing with an arbitrary renunciation of a more detailed analysis of atomic phenomena, but with a recognition that such an analysis is to principle excluded." (Niels Bohr, "Atomic Theory and Human Knowledge", 1958)

"It is, indeed, perhaps the greatest prospect of humanistic studies to contribute through an increasing knowledge of the history of cultural development to that gradual removal of prejudices which is the common aim of all science." (Niels Bohr, "Atomic Physics and Human Knowledge", 1958)

"Physics is to be regarded not so much as the study of something a priori given, but rather as the development of methods of ordering and surveying human experience. In this respect our task must be to account for such experience in a manner independent of individual subjective judgement and therefore objective in the sense that it can be unambiguously communicated in ordinary human language." (Niels Bohr, "The Unity of Human Knowledge", 1960)

"Every sentence I utter must be understood not as an affirmation, but as a question." (Niels Bohr)

"How wonderful that we have met with a paradox. Now we have some hope of making progress." (Niels Bohr)

"It is wrong to think that the task of physics is to find out how Nature is. Physics concerns what we can say about Nature." (Niels Bohr)

"The opposite of a correct statement is a false statement. The opposite of a profound truth may well be another profound truth." (Niels Bohr)

"Those who are not shocked when they first come across quantum mechanics cannot possibly have understood it." (Niels Bohr)

"We are trapped by language to such a degree that every attempt to formulate insight is a play on words." (Niels Bohr)

Eugene P Wigner - Collected Quotes

"The regularities in the phenomena which physical science endeavors to uncover are called the laws of nature. The name is actually very appropriate. Just as legal laws regulate actions and behavior under certain conditions but do not try to regulate all action and behavior, the laws of physics also determine the behavior of its objects of interest only under certain well-defined conditions but leave much freedom otherwise." (Eugene P Wigner, "Events, Laws of Nature, and Invariance principles", [Nobel lecture] 1914)

"The simplicities of natural laws arise through the complexities of the languages we use for their expression." (Eugene P Wigner, 1959)

"Nothing in our experience suggests the introduction of [complex numbers]. Indeed, if a mathematician is asked to justify his interest in complex numbers, he will point, with some indignation, to the many beautiful theorems in the theory of equations, of power series, and of analytic functions in general, which owe their origin to the introduction of complex numbers. The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius." (Eugene P Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", Communications in Pure and Applied Mathematics 13 (1), 1960)

"Somebody once said that philosophy is the misuse a terminology which was invented just for this purpose. In the same vein, I would say that mathematics is the science of skillful operations with concepts and rules invented just for this purpose."  (Eugene Wigner, "The of Mathematics in the Natural Sciences," Communications on Pure Applied Mathematics 13 (2), 1960)

"The enormous usefulness of mathematics in natural sciences is something bordering on the mysterious, and there is no rational explanation for it. It is not at all natural that ‘laws of nature’ exist, much less that man is able to discover them. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning." (Eugene P Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," 1960)

"The mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language." (Eugene P Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", Communications in Pure and Applied Mathematics 13 (1), 1960)

"[…] mathematics is the science of skillful operations with concepts and rules invented just for this purpose." (Eugene P Wigner, 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences," 1960)

"The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius." (Eugene P Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", Communications in Pure and Applied Mathematics 13 (1), 1960)

"Physics does not endeavour to explain nature. In fact, the great success of physics is due to a restriction of its objectives: it only endeavours to explain the regularities in the behavior of objects." (Eugene P Wigner, "Events, Laws of Nature, and Invariance Principles", [Nobel Lecture], 1963)

"We have ceased to expect from physics an explanation of all events, even of the gross structure of the universe, and we aim only at the discovery of the laws of nature, that is the regularities, of the events." (Eugene P Wigner, "Events, Laws of Nature, and Invariance Principles", [Nobel Lecture], 1963)

"We know many laws of nature and we hope and expect to discover more. Nobody can foresee the next such law that will be discovered. Nevertheless, there is a structure in laws of nature which we call the laws of invariance. This structure is so far-reaching in some cases that laws of nature were guessed on the basis of the postulate that they fit into the invariance structure." (Eugene P Wigner, "The Role of Invariance Principles in Natural Philosophy", 1963)

“Physics can teach us only what the laws of nature are today. It is only Astronomy that can teach us what the initial conditions for these laws are.” (Eugene P Wigner, “The Case for Astronomy”, Proceedings of the American Philosophical Society Vol. 8 (1), 1964)

"It is now natural for us to try to derive the laws of nature and to test their validity by means of the laws of invariance, rather than to derive the laws of invariance from what we believe to be the laws of nature." (Eugene P Wigner, "Symmetries and Reflections: Scientific Essays", 1967)

"I believe that the present laws of physics are at least incomplete without a translation into terms of mental phenomena." (Eugene P Wigner, "Physics and the Explanation of Life", 1970)

"In science, it is not speed that is the most important. It is the dedication, the commitment, the interest and the will to know something and to understand it - these are the things that come first." (Eugene P Wigner, [interview by István Kardos] 1978)

"Part of the art and skill of the engineer and of the experimental physicist is to create conditions in which certain events are sure to occur." (Eugene P Wigner, "Symmetries and Reflections", 1979)

"Physics is becoming so unbelievably complex that it is taking longer and longer to train a physicist. It is taking so long, in fact, to train a physicist to the place where he understands the nature of physical problems that he is already too old to solve them." (Eugene P Wigner) 

"The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation of it." (Eugene P Wigner) 

 

"With thermodynamics, one can calculate almost everything crudely; with kinetic theory, one can calculate fewer things, but more accurately; and with statistical mechanics, one can calculate almost nothing exactly." (Eugene P Wigner) 

Charles Babbage - Collected Quotes

"For one person who is blessed with the power of invention, many will always be found who have the capacity of applying principles." (Charles Babbage, "Reflections on the Decline of Science in England, and on Some of Its Causes", 1830)

"It can happen to but few philosophers, and but at distant intervals, to snatch a science, like Dalton, from the chaos of indefinite combination, and binding it in the chains of number, to exalt it to rank amongst the exact. Triumphs like these are necessarily 'few and far between’." (Charles Babbage, "Reflections on the Decline of Science in England, and on Some of Its Causes", 1830)

"The first steps in the path of discovery, and the first approximate measures, are those which add most to the existing knowledge of mankind." (Charles Babbage, "Reflections on the Decline of Science in England", 1830)

"Science and knowledge are subject, in their extension and increase, to laws quite opposite to those which regulate the material world. Unlike the forces of molecular attraction, which cease at sensible distances; or that of gravity, which decreases rapidly with the increasing distance from the point of its origin; the farther we advance from the origin of our knowledge, the larger it becomes, and the greater power it bestows upon its cultivators, to add new fields to its dominions." (Charles Babbage, "On the Economy of Machinery and Manufactures", 1832)

"The errors which arise from the absence of facts are far more numerous and more durable than those which result from unsound reasoning respecting true data." (Charles Babbage, "On the Economy of Machinery and Manufactures", 1832)

"Remember that accumulated knowledge, like accumulated capital, increases at compound interest: but it differs from the accumulation of capital in this; that the increase of knowledge produces a more rapid rate of progress, whilst the accumulation of capital leads to a lower rate of interest. Capital thus checks its own accumulation: knowledge thus accelerates its own advance. Each generation, therefore, to deserve comparison with its predecessor, is bound to add much more largely to the common stock than that which it immediately succeeds." (Charles Babbage, "The Exposition of 1851: Or the Views of Industry, Science and Government of England", 1851)

"Mechanical Notation […] I look upon it as one of the most important additions I have made to human knowledge. It has placed the construction of machinery in the rank of a demonstrative science. The day will arrive when no school of mechanical drawing will be thought complete without teaching it." (Charles Babbage, "Passages From the Life of a Philosopher", 1864)

"The more man inquires into the laws which regulate the material universe, the more he is convinced that all its varied forms arise from the action of a few simple principles. These principles themselves converge, with accelerating force, towards some still more comprehensive law to which all matter seems to be submitted. Simple as that law may possibly be, it must be remembered that it is only one amongst an infinite number of simple laws: that each of these laws has consequences at least as extensive as the existing one, and therefore that the Creator who selected the present law must have foreseen the consequences of all other laws." (Charles Babbage, "Passages From the Life of a Philosopher", 1864)

"The whole of the developments and operations of analysis are now capable of being executed by machinery. […] As soon as an Analytical Engine exists, it will necessarily guide the future course of science." (Charles Babbage, "Passages from the Life of a Philosopher", 1864)

"Whenever a man can get hold of numbers, they are invaluable: if correct, they assist in informing his own mind, but they are still more useful in deluding the minds of others. Numbers are the masters of the weak, but the slaves of the strong." (Charles Babbage, "Passages from the Life of a Philosopher", 1864)

"I wish to God these calculations had been executed by steam." (Charles Babbage)

Niels H Abel - Collected Quotes

"The mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it. However, if I am not mistaken, they have not as yet succeeded. I therefore dare hope that the mathematicians will receive this memoir with good will, for its purpose is to fill this gap in the theory of algebraic equations." (Niels H Abel, "Memoir on algebraic equations, proving the impossibility of a solution of the general equation of the fifth degree", 1824)

"I shall devote all my efforts to bring light into the immense obscurity that today reigns in Analysis. It so lacks any plan or system, that one is really astonished that so many people devote themselves to it - and, still worse, it is absolutely devoid of any rigour." (Niels H Abel, "Oeuvres", 1826)

"In analysis, one is largely occupied by functions which can be expressed as powers. As soon as other powers enter - this, however, is not often the case - then it does not work any more and a number of connected, incorrect theorems arise from false conclusions." (Niels H Abel, [Letter to Christoffer Hansteen] 1826) 

"If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without a foundation." (Niels H Abel) 

"It appears to me that if one wishes to make progress in mathematics, one should study the masters and not the pupils." (Niels H Abel)

"The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes." (Niels H Abel) 

"With the exception of the geometric series, there does not exist in all of mathematics a single infinite series whose sum has been determined rigorously." (Niels H Abel)

Quotes about Niels H Abel:

"All of Abel's works carry the imprint of an ingenuity and force of thought which is unusual and sometimes amazing, even if the youth of the author is not taken into consideration. One may say that he was able to penetrate all obstacles down to the very foundations of the problems, with a force which appeared irresistible; he attacked the problems with extraordinary energy; he regarded them from above and was able to soar so high over their present state that all difficulties seemed to vanish under the victorious onslaught of his genius. [...] But it was not only his great talent which created the respect for Abel and made his loss infinitely regrettable. He distinguished himself equally by the purity and nobility of his character and by a rare modesty which made his person cherished to the same unusual degree as was his genius." (August L Crelle, "Crelle's Journal", 1841) 

Heinrich Heine - Collected Quotes

"Nothing is more futile than theorizing about music. No doubt there are laws, mathematically strict laws, but these laws are not music; they are only its conditions […] The essence of music is revelation." (Heinrich Heine, "Letters on the French Stage", 1837)

"What then is music? […] It exists between thought and phenomenon, like a twilight medium, it stands between spirit and matter, related to and yet different from both; it is spirit, but spirit governed by time; it is matter, but matter that can manage without space." (Heinrich Heine, "On the French Stage", 1837)

"Great genius takes shape by contact with another great genius, but less by assimilation than by friction."  (Heinrich Heine)

"Like a great poet, Nature produces the greatest results with the simplest means." (Heinrich Heine)

D'Arcy W Thompson - Collected Quotes

"It behooves us always to remember that in physics it has taken great men to discover simple things." (D'Arcy W Thompson, "On Growth and Form", 1951)

"Numerical precision is the very soul of science." (D'Arcy W Thompson, "On Growth and Form", 1951)

"The concept of an average, the equation to a curve, the description of a froth or cellular tissue, all come within the scope of mathematics for no other reason than that they are summations of more elementary principles or phenomena. Growth and Form are throughout of this composite nature; therefore the laws of mathematics are bound to underlie them, and her methods to be peculiarly fitted to interpret them." (D'Arcy W Thompson, "On Growth and Form", 1951)

"The harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty." (Sir D’Arcy W Thompson, "On Growth and Form", 1951)

"The rate of growth deserves to be studied as a necessary preliminary to the theoretical study of form, and organic form itself is found, mathematically speaking, to be a function of time. […] We might call the form of an organism an event in space-time, and not merely a configuration in space."  (Sir D’Arcy W Thompson, "On Growth and Form", 1951)

"The waves of the sea, the little ripples on the shore, the sweeping curve of the sandy bay between the headlands, the outline of the hills, the shape of the clouds, all these are so many riddles of form, so many problems of morphology." (Sir D’Arcy W Thompson, "On Growth and Form", 1951)

"To seek not for end but for antecedents is the way of the physicist, who finds "causes" in what he has learned to recognise as fundamental properties, or inseparable concomitants, or unchanging laws, of matter and of energy." (Sir D’Arcy W Thompson, "On Growth and Form", 1951)

Martin Heidegger - Collected Quotes

"The average, vague understanding of being can be permeated by traditional theories and opinions about being in such a way that these theories, as the sources of the prevailing understanding, remain hidden." (Martin Heidegger, "Being and Time", 1927)

"Philosophy is metaphysics. Metaphysics thinks beings as a whole - the world, man, God - with respect to Being, with respect to the belonging together of beings in Being. Metaphysics thinks beings as being in the manner of representational thinking which gives reasons." (Martin Heidegger, "The End of Philosophy and the Task of Thinking", 1964)

"All great insights and discoveries are not only usually thought by several people at the same time, they must also be re-thought in that unique effort to truly say the same thing about the same thing." (Martin Heidegger, "What Is A Thing", 1967)

"Our expression 'the mathematical' always has two meanings. It means, first, what can be learned in the manner we have indicated, and only in that way, and, second, the manner of learning and the process itself. The mathematical is that evident aspect of things within which we are always already moving and according to which we experience them as things at all, and as such things. The mathematical is this fundamental position we take toward things by which we take up things as already given to us, and as they must and should be given. Therefore, the mathematical is the fundamental presupposition of the knowledge of things." (Martin Heidegger, "What Is A Thing", 1967)

"The mathematical is that evident aspect of things within which we are always already moving and according to which we experience them as things at all, and as such things. The mathematical is this fundamental position we take toward things by which we take up things as already given to us, and as they must and should be given. Therefore, the mathematical is the fundamental presupposition of the knowledge of things." (Martin Heidegger, "Modern Science, Metaphysics and Mathematics", 1967)

"We take cognizance of all this and learn it without regard for the things. Numbers are the most familiar form of the mathematical because, in our usual dealing with things, when we calculate or count, numbers are the closest to that which we recognize in things without deriving it from them. For this reason numbers are the most familiar form of the mathematical. In this way, this most familiar mathematical becomes mathematics. But the essence of the mathematical does not lie in number as purely delimiting the pure ‘how much’, but vice versa. Because number has such a nature, therefore, it belongs to the learnable in the sense of mathesis." (Martin Heidegger, "Modern Science, Metaphysics and Mathematics", 1967)

"Teaching is more difficult than learning because what teaching calls for is this: to let learn. The real teacher, in fact, let nothing else be learned than learning. His conduct, therefore, often produces the impression that we properly learn nothing from him, if by ‘learning’ we now suddenly understand merely the procurement of useful information." (Martin Heidegger, "What is called thinking?", 1968)

Thomas Browne - Collected Quotes

"Every man is not a proper champion for truth, nor fit to take up the gauntlet in the cause of verity: many from the ignorance of these maxims, and an inconsiderate zeal for truth, have too rashly charged the troops of error, and remain as trophies unto the enemies of truth. A man may be in as just possession of truth as of a city, and yet be forced to surrender: ’tis therefore far better to enjoy her with peace than to hazard her on a battle: if therefore there rise any doubts in my way, I do forget them, or at least defer them, till my better settled judgment and more manly reason be able to resolve them." (Sir Thomas Browne, "Religio Medici", 1643)

"Nature is not at variance with art nor art with nature, they both being the servants of his providence: art is the perfection of nature." (Sir Thomas Browne," Religio Medici", 1643)

"To believe only in possibilities, is not faith, but mere Philosophy." (Sir Thomas Browne," Religio Medici", 1643)

"Knowledge is made by oblivion, and to purchase a clear and warrantable body of truth, we must forget and part with much we know." (Sir Thomas Browne, "Pseudodoxia Epidemica", 1646)

"All things began in order, so shall they end, and so shall they begin again; according to the ordainer of order and mystical mathematics of the city of heaven." (Sir Thomas Browne, "The Garden of Cyrus", 1658)

"Natura nihil agit frustra [Nature does nothing in vain] is the only indisputable axiom in philosophy. There are no grotesques in nature; not any thing framed to fill up empty cantons, and unncecessary spaces." (Thomas Browne)

Roger Bacon - Collected Quotes

"All science requires mathematics […] the knowledge of mathematical things is almost innate in us […] this is the easiest of sciences. A fact which is obvious in that no one’s brain rejects it. For laymen and people who are utterly illiterate know how to count and reckon.” (Roger Bacon, "Opus Majus”, 1267)

"For the things of this world cannot be made known without a knowledge of mathematics.” (Roger Bacon, "Opus Majus”, 1267)

"If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.” (Roger Bacon, "Opus Majus", 1267)

"Mathematics is the door and key to the sciences.” (Roger Bacon, "Opus Majus", 1267)

"Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain, nor does it remove doubt so that the mind may rest on the intuition of truth, unless the mind discovers it by the path of experience.” (Roger Bacon, "Opus Majus", 1267)

"There are four great sciences, without which the other sciences cannot be known nor a knowledge of things secured […] Of these sciences the gate and key is mathematics […] He who is ignorant of this [mathematics] cannot know the other sciences nor the affairs of this world.” (Roger Bacon, "Opus Majus", 1267)

"There are two modes of acquiring knowledge, namely, by reasoning and experience. Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain, nor does it remove doubt so that the mind may rest on the intuition of truth unless the mind discovers it by the path of experience.” (Roger Bacon, "Opus Majus", 1267)

"Without experience nothing can be sufficiently known. For there are two modes of acquiring knowledge, namely, by reasoning and by experience. Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain [...] unless the mind discovers it by the method of experience." (Roger Bacon, "Opus Majus", 1267)

"All sciences are connected; they lend each other material aid as parts of one great whole, each doing its own work, not for itself alone, but for the other parts; as the eye guides the body and the foot sustains it and leads it from place to place.” (Roger Bacon, "Opus Tertium”, [1266–1268])

"Neglect of mathematics works injury to all knowledge, since one who is ignorant of it cannot know the other sciences of the things of this world. And what is worst, those who are thus ignorant are unable to perceive their own ignorance and so do not seek a remedy." (Roger Bacon)

"The strongest arguments prove nothing so long as the conclusions are not verified by experience. Experimental science is the queen of sciences and the goal of all speculation." (Roger Bacon)

Jean le Rond d’Alembert - Collected Quotes

"Day by day natural science accumulates new riches […] The true system of the World has been recognized, developed and perfected […] Everything has been discussed and analyzed, or at least mentioned.” (Jean le Rond d’Alembert, "Elements of Philosophy", 1759)

"One must admit that it is not a simple matter to accurately outline the idea of negative numbers, and that some capable people have added to the confusion by their inexact pronouncements. To say that the negative numbers are below nothing is to assert an unimaginable thing.” (Jean le Rond d'Alembert, "Negatif”, Encyclopédie [1751 – 1772])

"[…] the algebraic rules of operation with negative numbers are generally admitted by everyone and acknowledged as exact, whatever idea we may have about this quantities. " (Jean le Rond d'Alembert, Encyclopédie, [1751 – 1772])

"Thus, metaphysics and mathematics are, among all the sciences that belong to reason, those in which imagination has the greatest role.” (Jean le Rond d'Alembert, Encyclopédie, [1751 – 1772])

"Geometrical truths are in a way asymptotes to physical truths, that is to say, the latter approach the former indefinitely near without ever reaching them exactly.” (Jean le Rond d’Alembert)

"The imagination in a mathematician who creates makes no less difference than in a poet who invents […]." (Jean le Rond d'Alembert, Encyclopedie, [1751 – 1772])

"To someone who could grasp the universe from one unified viewpoint, the entire creation would appear as a unique fact and a great truth.” (Jean le Rond d'Alembert)

"We shall content ourselves with the remark that if mathematics (as is asserted with sufficient reason) only make straight the minds which are without bias, so they only dry up and chill the minds already prepared for this operation by nature.” (Jean le Rond d'Alembert)

Mental Models XX

“We perceive an image of truth, and possess only a lie.” (Blaise Pascal, “Pensées”, 1670)

"Our mental vision or conception of ideas is nothing but a revelation made to us by our Maker. When we voluntarily turn our thoughts to any object, and raise up its image in the fancy, it is not the will which creates that idea: It is the universal Creator, who discovers it to the mind, and renders it present to us.” (David Hume, “An Enquiry Concerning Human Understanding”, 1748)

"Wit, you know, is the unexpected copulation of ideas, the discovery of some occult relation between images in appearance remote from each other." (Samuel Johnson, "The Rambler", 1750)

“This schematism of our understanding, in its application to appearances and their mere form, is an art concealed in the depths of the human soul, whose real modes of activity nature is hardly likely ever to allow us to discover, and to have open to our gaze.” (Immanuel Kant, “Critique of Pure Reason”, 1781) 

“Everything possible to be believed is an image of truth.” (William Blake, “The Marriage of Heaven and Hell”, 1790)

"The identifying ourselves with the visual image of ourselves has become an instinct; the habit is already old. The picture of me, the me that is seen, is me." (David H Lawrence, "Art and Morality", 1925)

“We do not know why the imagination has accepted that image before the reason can reject it; or why such correspondences seem really to correspond to something in the soul.” (Gilbert K Chesterton, “The Everlasting Man”, 1925)

“If we hang beautiful pictures on the walls of our souls, mental images that establish us in the habitual companionship of the highest that we know, and live with them long enough, we cannot will evil.” (Harry E Fosdick, “The Hope of the World”, 1933)

“Truth is a totality, the sum of many overlapping partial images. History, on the other hand, sacrifices totality in the interest of continuity.” (Edmund Leach, "Brain-Twister”, 1967)

Edward R Tufte - Collected Quotes

"Almost all efforts at data analysis seek, at some point, to generalize the results and extend the reach of the conclusions beyond a particular set of data. The inferential leap may be from past experiences to future ones, from a sample of a population to the whole population, or from a narrow range of a variable to a wider range. The real difficulty is in deciding when the extrapolation beyond the range of the variables is warranted and when it is merely naive. As usual, it is largely a matter of substantive judgment - or, as it is sometimes more delicately put, a matter of 'a priori nonstatistical considerations'." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

"If two or more describing variables in an analysis are highly intercorrelated, it will be difficult and perhaps impossible to assess accurately their independent impacts on the response variable. As the association between two or more describing variables grows stronger, it becomes more and more difficult to tell one variable from the other. This problem, called "multicollinearity" in the statistical jargon, sometimes causes difficulties in the analysis of nonexperimental data. […] No statistical technique can go very far to remedy the problem because the fault lies basically with the data rather than the method of analysis. Multicollinearity weakens inferences based on any statistical method--regression, path analysis, causal modeling, or cross-tabulations (where the difficulty shows up as a lack of deviant cases and as near-empty cells)."  (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

"[…] it is not enough to say: 'There's error in the data and therefore the study must be terribly dubious'. A good critic and data analyst must do more: he or she must also show how the error in the measurement or the analysis affects the inferences made on the basis of that data and analysis." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

"Our inability to measure important factors does not mean either that we should sweep those factors under the rug or that we should give them all the weight in a decision. Some important factors in some problems can be assessed quantitatively. And even though thoughtful and imaginative efforts have sometimes turned the 'unmeasurable' into a useful number, some important factors are simply not measurable. As always, every bit of the investigator's ingenuity and good judgment must be brought into play. And, whatever un- knowns may remain, the analysis of quantitative data nonetheless can help us learn something about the world - even if it is not the whole story." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

"Random data contain no substantive effects; thus if the analysis of the random data results in some sort of effect, then we know that the analysis is producing that spurious effect, and we must be on the lookout for such artifacts when the genuine data are analyzed." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

"Typically, data analysis is messy, and little details clutter it. Not only confounding factors, but also deviant cases, minor problems in measurement, and ambiguous results lead to frustration and discouragement, so that more data are collected than analyzed. Neglecting or hiding the messy details of the data reduces the researcher's chances of discovering something new." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

"The use of statistical methods to analyze data does not make a study any more 'scientific', 'rigorous', or 'objective'. The purpose of quantitative analysis is not to sanctify a set of findings. Unfortunately, some studies, in the words of one critic, 'use statistics as a drunk uses a street lamp, for support rather than illumination'. Quantitative techniques will be more likely to illuminate if the data analyst is guided in methodological choices by a substantive understanding of the problem he or she is trying to learn about. Good procedures in data analysis involve techniques that help to (a) answer the substantive questions at hand, (b) squeeze all the relevant information out of the data, and (c) learn something new about the world." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

"Graphical excellence is that which gives to the viewer the greatest number of ideas in the shortest time with the least ink in the smallest space." (Edward R Tufte, "The Visual Display of Quantitative Information", 1983)

"Inept graphics also flourish because many graphic artists believe that statistics are boring and tedious. It then follows that decorated graphics must pep up, animate, and all too often exaggerate what evidence there is in the data. […] If the statistics are boring, then you've got the wrong numbers." (Edward R Tufte, "The Visual Display of Quantitative Information", 1983)

"Of course statistical graphics, just like statistical calculations, are only as good as what goes into them. An ill-specified or preposterous model or a puny data set cannot be rescued by a graphic (or by calculation), no matter how clever or fancy. A silly theory means a silly graphic." (Edward R Tufte, "The Visual Display of Quantitative Information", 1983)

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

"Vigorous writing is concise. A sentence should contain no  unnecessary words, a paragraph no unnecessary sentences, for the same reason that a drawing should have no  unnecessary lines and a machine no unnecessary parts. This requires not that the writer make all his sentences short, or that he avoid all detail and treat his subjects only in outline,  but that every word tell." (Edward Tufte, "The Visual Display of Quantitative Information", 1983)

"What about confusing clutter? Information overload? Doesn't data have to be ‘boiled down’ and  ‘simplified’? These common questions miss the point, for the quantity of detail is an issue completely separate from the difficulty of reading. Clutter and confusion are failures of design, not attributes of information." (Edward R Tufte, "Envisioning Information", 1990)

"Audience boredom is usually a content failure, not a decoration failure." (Edward R Tufte, "The cognitive style of PowerPoint", 2003)

"If your words or images are not on point, making them dance in color won't make them relevant." (Edward R Tufte, "The cognitive style of PowerPoint", 2003)

John W Tukey - Collected Quotes

"[We] need men who can practice science - not a particular science - in a word, we need scientific generalists." (John W Tukey, "The Education of a Scientific Generalist", 1949)

"[...] the whole of modern statistics, philosophy and methods alike, is based on the principle of interpreting what did happen in terms of what might have happened." (John W Tukey, "Standard Methods of Analyzing Data, 1951)

"Just remember that not all statistics has been mathematized - and that we may not have to wait for its mathematization in order to use it." (John W Tukey, "The Growth of Experimental Design in a Research Laboratory, 1953)

"Difficulties in identifying problems have delayed statistics far more than difficulties in solving problems." (John W Tukey, Unsolved Problems of Experimental Statistics, 1954)

"Predictions, prophecies, and perhaps even guidance - those who suggested this title to me must have hoped for such-even though occasional indulgences in such actions by statisticians has undoubtedly contributed to the characterization of a statistician as a man who draws straight lines from insufficient data to foregone conclusions!" (John W Tukey, "Where do We Go From Here?", Journal of the American Statistical Association, Vol. 55 (289), 1960)

"Today one of statistics' great needs is a body of able investigators who make it clear to the intellectual world that they are scientific statisticians. and they are proud of that fact that to them mathematics is incidental, though perhaps indispensable." (John W Tukey, "Statistical and Quantitative Methodology, 1961)

"If data analysis is to be well done, much of it must be a matter of judgment, and ‘theory’ whether statistical or non-statistical, will have to guide, not command." (John W Tukey, "The Future of Data Analysis", Annals of Mathematical Statistics, Vol. 33 (1), 1962)

"The most important maxim for data analysis to heed, and one which many statisticians seem to have shunned is this: ‘Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise.’ Data analysis must progress by approximate answers, at best, since its knowledge of what the problem really is will at best be approximate." (John W Tukey, "The Future of Data Analysis", Annals of Mathematical Statistics, Vol. 33, No. 1, 1962)

"The histogram, with its columns of area proportional to number, like the bar graph, is one of the most classical of statistical graphs. Its combination with a fitted bell-shaped curve has been common since the days when the Gaussian curve entered statistics. Yet as a graphical technique it really performs quite poorly. Who is there among us who can look at a histogram-fitted Gaussian combination and tell us, reliably, whether the fit is excellent, neutral, or poor? Who can tell us, when the fit is poor, of what the poorness consists? Yet these are just the sort of questions that a good graphical technique should answer at least approximately." (John W Tukey, "The Future of Processes of Data Analysis", 1965)

"The first step in data analysis is often an omnibus step. We dare not expect otherwise, but we equally dare not forget that this step, and that step, and other step, are all omnibus steps and that we owe the users of such techniques a deep and important obligation to develop ways, often varied and competitive, of replacing omnibus procedures by ones that are more sharply focused." (John W Tukey, "The Future of Processes of Data Analysis", 1965)

"The basic general intent of data analysis is simply stated: to seek through a body of data for interesting relationships and information and to exhibit the results in such a way as to make them recognizable to the data analyzer and recordable for posterity. Its creative task is to be productively descriptive, with as much attention as possible to previous knowledge, and thus to contribute to the mysterious process called insight." (John W Tukey & Martin B Wilk, "Data Analysis and Statistics: An Expository Overview", 1966)

"Comparable objectives in data analysis are (l) to achieve more specific description of what is loosely known or suspected; (2) to find unanticipated aspects in the data, and to suggest unthought-of-models for the data's summarization and exposure; (3) to employ the data to assess the (always incomplete) adequacy of a contemplated model; (4) to provide both incentives and guidance for further analysis of the data; and (5) to keep the investigator usefully stimulated while he absorbs the feeling of his data and considers what to do next." (John W Tukey & Martin B Wilk, "Data Analysis and Statistics: An Expository Overview", 1966)

"The science and art of data analysis concerns the process of learning from quantitative records of experience. By its very nature it exists in relation to people. Thus, the techniques and the technology of data analysis must be harnessed to suit human requirements and talents. Some implications for effective data analysis are: (1) that it is essential to have convenience of interaction of people and intermediate results and (2) that at all stages of data analysis the nature and detail of output, both actual and potential, need to be matched to the capabilities of the people who use it and want it." (John W Tukey & Martin B Wilk, "Data Analysis and Statistics: An Expository Overview", 1966)

"In many instances, a picture is indeed worth a thousand words. To make this true in more diverse circumstances, much more creative effort is needed to pictorialize the output from data analysis. Naive pictures are often extremely helpful, but more sophisticated pictures can be both simple and even more informative." (John W Tukey & Martin B Wilk, "Data Analysis and Statistics: An Expository Overview", 1966)

"Data analysis must be iterative to be effective. [...] The iterative and interactive interplay of summarizing by fit and exposing by residuals is vital to effective data analysis. Summarizing and exposing are complementary and pervasive." (John W Tukey & Martin B Wilk, "Data Analysis and Statistics: An Expository Overview", 1966)

"Summarizing data is a process of constrained and partial a process that essentially and inevitably corresponds to description - some sort of fitting, though it need not necessarily involve formal criteria or well-defined computations." (John W Tukey & Martin B Wilk, "Data Analysis and Statistics: An Expository Overview", 1966)

"The typical statistician has learned from bitter experience that negative results are just as important as positive ones, sometimes more so." (John W Tukey, "A Statistician's Comment", 1967)

"It is fair to say that statistics has made its greatest progress by having to move away from certainty [...] If we really want to make progress, we need to identify our next step away from certainty." (John W Tukey, "What Have Statisticians Been Forgetting", 1967)

"Every student of the art of data analysis repeatedly needs to build upon his previous statistical knowledge and to reform that foundation through fresh insights and emphasis." (John W Tukey, "Data Analysis, Including Statistics", 1968)

"Every graph is at least an indication, by contrast with some common instances of numbers." (John W Tukey, "Data Analysis, Including Statistics", 1968)

"Nothing can substitute for relatively direct assessment of variability." (John W Tukey, "Data Analysis, Including Statistics", 1968)

"No one knows how to appraise a procedure safely except by using different bodies of data from those that determined it."  (John W Tukey, "Data Analysis, Including Statistics", 1968)

"The problems of different fields are much more alike than their practitioners think, much more alike than different." (John W Tukey, "Analyzing Data: Sanctification or Detective Work?", 1969)

"[...] bending the question to fit the analysis is to be shunned at all costs." (John W Tukey, "Analyzing Data: Sanctification or Detective Work?", 1969)

"Data analysis is in important ways an antithesis of pure mathematics." (John W Tukey, "Data Analysis, Computation and Mathematics", 1972)

"Undoubtedly, the swing to exploratory data analysis will go somewhat too far. However : It is better to ride a damped pendulum than to be stuck in the mud." (John W Tukey, "Exploratory Data Analysis as Part of a Larger Whole", 1973)

"The greatest value of a picture is when it forces us to notice what we never expected to see." (John W Tukey, "Exploratory Data Analysis", 1977)

"[...] exploratory data analysis is an attitude, a state of flexibility, a willingness to look for those things that we believe are not there, as well as for those we believe might be there. Except for its emphasis on graphs, its tools are secondary to its purpose." (John W Tukey, [comment] 1979)

"There is NO question of teaching confirmatory OR exploratory - we need to teach both." (John W Tukey, "We Need Both Exploratory and Confirmatory", 1980)

"Finding the question is often more important than finding the answer." (John W Tukey, "We Need Both Exploratory and Confirmatory", 1980)

"[...] any hope that we are smart enough to find even transiently optimum solutions to our data analysis problems is doomed to failure, and, indeed, if taken seriously, will mislead us in the allocation of effort, thus wasting both intellectual and computational effort." (John W Tukey, "Choosing Techniques for the Analysis of Data", 1981)

"Detailed study of the quality of data sources is an essential part of applied work. [...] Data analysts need to understand more about the measurement processes through which their data come. To know the name by which a column of figures is headed is far from being enough." (John W Tukey, "An Overview of Techniques of Data Analysis, Emphasizing Its Exploratory Aspects", 1982)

"Exploratory data analysis, EDA, calls for a relatively free hand in exploring the data, together with dual obligations: (•) to look for all plausible alternatives and oddities - and a few implausible ones, (graphic techniques can be most helpful here) and (•) to remove each appearance that seems large enough to be meaningful - ordinarily by some form of fitting, adjustment, or standardization [...] so that what remains, the residuals, can be examined for further appearances." (John W Tukey, "Introduction to Styles of Data Analysis Techniques", 1982)

"The combination of some data and an aching desire for an answer does not ensure that a reasonable answer can be extracted from a given body of data." (John W Tukey, "Sunset Salvo", The American Statistician Vol. 40 (1), 1986)

"The worst, i.e., most dangerous, feature of 'accepting the null hypothesis' is the giving up of explicit uncertainty. […] Mathematics can sometimes be put in such black-and-white terms, but our knowledge or belief about the external world never can." (John W Tukey, "The Philosophy of Multiple Comparisons", Statistical Science Vol. 6 (1), 1991)

"Statistics is the science, the art, the philosophy, and the technique of making inferences from the particular to the general." (John W Tukey)

Mental Models XIX

"'Schema' refers to an active organisation of past reactions, or of past experiences, which must always be supposed to be operating in any well-adapted organic response. That is, whenever there is any order or regularity of behavior, a particular response is possible only because it is related to other similar responses which have been serially organised, yet which operate, not simply as individual members coming one after another, but as a unitary mass. Determination by schemata is the most fundamental of all the ways in which we can be influenced by reactions and experiences which occurred some time in the past. All incoming impulses of a certain kind, or mode, go together to build up an active, organised setting: visual, auditory, various types of cutaneous impulses and the like, at a relatively low level; all the experiences connected by a common interest: in sport, in literature, history, art, science, philosophy, and so on, on a higher level." (Frederic C Bartlett, "Remembering: A study in experimental and social psychology", 1932)

"[T]he sudden inventions characteristic of the sixth stage [of infant development] are in reality the product of a long evolution of schemata and not only of an internal maturation of perceptive structures. [..] This is revealed by the existence of a fifth stage, characterized by experimental groping. […] What does this mean if not that the practice of actual experience is necessary in order to acquire the practice of mental experience and that invention does not arise entirely preformed despite appearances? (Jean Piaget, "The origin of intelligence in children" 1936)

"My hypothesis then is that thought models, or parallels, reality - that its essential feature is not ‘the mind’, ‘the self’, ‘sense-data’, nor propositions but symbolism, and that this symbolism is largely of the same kind as that which is familiar to us in mechanical devices which aid thought and calculation." (Kenneth Craik, "The Nature of Explanation", 1943)

"A person is changed by the contingencies of reinforcement under which he behaves; he does not store the contingencies. In particular, he does not store copies of the stimuli which have played a part in the contingencies. There are no 'iconic representations' in his mind; there are no 'data structures stored in his memory'; he has no 'cognitive map' of the world in which he has lived. He has simply been changed in such a way that stimuli now control particular kinds of perceptual behavior." (Burrhus F Skinner, "About behaviorism", 1974)

"Imagining is not perceiving, but images are indeed derivatives of perceptual activity. In particular, they are the anticipatory phases of that activity, schemata that the perceiver has detached from the perceptual cycle for other purposes. […] The experience of having an image is just the inner aspect of a readiness to perceive the imagined object. (Ulrich Neisser, "Cognition and Reality" 1976)

"[I]t seems (to many) that we cannot account for perception unless we suppose it provides us with an internal image (or model or map) of the external world, and yet what good would that image do us unless we have an inner eye to perceive it, and how are we to explain its capacity for perception? It also seems (to many) that understanding a heard sentence must be somehow translating it into some internal message, but how will this message be understood: by translating it into something else? The problem is an old one, and let’s call it Hume’s Problem, for while he did not state it explicitly, he appreciated its force and strove mightily to escape its clutches. (Daniel Dennett, "Brainstorms: Philosophical essays on mind and psychology", 1978)

"A schema, then is a data structure for representing the generic concepts stored in memory. There are schemata representing our knowledge about all concepts; those underlying objects, situations, events, sequences of events, actions and sequences of actions. A schema contains, as part of its specification, the network of interrelations that is believed to normally hold among the constituents of the concept in question. A schema theory embodies a prototype theory of meaning. That is, inasmuch as a schema underlying a concept stored in memory corresponds to the meaning of that concept, meanings are encoded in terms of the typical or normal situations or events that instantiate that concept." (David E Rumelhart, "Schemata: The building blocks of cognition", 1980)

"Once we have accepted a configuration of schemata, the schemata themselves provide a richness that goes far beyond our observations. […] In fact, once we have determined that a particular schema accounts for some event, we may not be able to determine which aspects of our beliefs are based on direct sensory information and which are merely consequences of our interpretation." (David E Rumelhart, "Schemata: The building blocks of cognition", 1980)

"Since mental models can take many forms and serve many purposes, their contents are very varied. They can contain nothing but tokens that represent individuals and identities between them, as in the sorts of models that are required for syllogistic reasoning. They can represent spatial relations between entities, and the temporal or causal relations between events. A rich imaginary model of the world can be used to compute the projective relations required for an image. Models have a content and form that fits them to their purpose, whether it be to explain, to predict, or to control." (Philip Johnson-Laird, "Mental models: Toward a cognitive science of language, inference, and consciousness", 1983)

"The basic idea is that schemata are data structures for representing the generic concepts stored in memory. There are schemata for generalized concepts underlying objects, situations, events, sequences of events, actions, and sequences of actions. Roughly, schemata are like models of the outside world. To process information with the use of a schema is to determine which model best fits the incoming information. Ultimately, consistent configurations of schemata are discovered which, in concert, offer the best account for the input. This configuration of schemata together constitutes the interpretation of the input. (David E Rumelhart, Paul Smolensky, James L McClelland & Geoffrey E Hinton, "Schemata and sequential thought processes in PDP models", 1986)

Gheorghe Ţiţeica - Collected Quotes

"At difficult times like this, the only salvation is an enthusiasm for science and elevated thinking, and from all sciences, mathematics, through its precise problems, through its rigorous proofs, gives the most important and immediate rewarding and serves then as a solid foundation for any other theoretical or applied profession." (Gheorghe Ţiţeica, "Gazeta Matematica", ["Mathematical Gazette"] XXXVI, 1916)

"Born at the same time with the Greek art, the mathematics kept in its canvas, in its intimate structure, a certain affinity with art. It comes to the same harmony in Euclid’s geometry as in the ancient temples. It is the same silence, the same balance in demonstrating a theorem as in the admirable columns of the Acropolis." (Gheorghe Ţiţeica)

"Mathematics is a way to express the natural laws, it is the simplest and the most appropriate mode to describe a general law or the flow of a phenomenon, it is the most perfect language for narrating a natural phenomenon." (Gheorghe Ţiţeica)

"The slyness, cunning, lie and how many other abilities that are employed, sometimes successfully, unfortunately even with very high success, in the everyday life, have no place in the mathematical proof." (Gheorghe Ţiţeica, "Mathematics and Art")

"The world of mathematics is an ideal world, governed by a crystal-like order and beauty."  (Gheorghe Ţiţeica, "Mathematics and Art")

"There exists, among mathematicians, a deep-seated and strong belief which sustains them in their abstract studies, namely that none of their problems can remain without any answer." (Gheorghe Ţiţeica)

Dan Barbilian - Collected Quotes

"As in geometry, I understand through poetry a particular symbolism for representing the possible forms of existence. [...] For me poetry is a prolongation of geometry, so that, while remaining a poet, I have never abandoned the divine domain of geometry." (Dan Barbilian, 1927)  

"I consider myself more of a practitioner of mathematics and less of a poet, and that only insofar as poetry recalls geometry. No matter how contradictory these two terms might seem at first sight, there is somewhere in the high realm of geometry a bright spot where it meets poetry." (Dan Barbilian, 1927) 

"Mathematical research can lend its organisational characteristics to poetry, whereby disjointed metaphors take on a universal sense. Similarly, the axiomatic foundations of group theory can be assimilated into a larger moral concept of a unified universe. Without this, mathematics would be a laborious Barbary." (Dan Barbilian, “The Autobiography of the Scientist”, 1940) 

“[…] the major mathematical research acquires an organization and orientation similar to the poetical function which, adjusting by means of metaphor disjunctive elements, displays a structure identical to the sensitive universe. Similarly, by means of its axiomatic or theoretical foundation, mathematics assimilates various doctrines and serves the instructive purpose, the one set up by the unifying moral universe of concepts. ” (Dan Barbilian, “The Autobiography of the Scientist”, 1940) 

"The domain of poetry is not the entire soul, but only that privileged part where echoes the playing of lyres. It is the place of all intelligible beauty: pure understanding, the honour of geometries." (Dan Barbilian, 1947)

"After all, Greek thought is expressed not only mythically, in fiction, but also directly, in theorems. The gate through which the Greek world may be discussed – and without the knowledge of which, in my opinion, one’s culture can not be deemed complete – is not necessarily Homer. Greek geometry is a wider gate, through which the eye might grasp an austere, yet essential landscape." (Dan Barbilian, 1967) 

" [if a poem] admits an explanation, rationally it admits an infinity. An exegesis can in no way be absolute. A poet provided certain mathematics can give not one, not two, but a great number of explanations of a hidden poem." (Dan Barbilian, 1968) 

"The mathematical works thrall and delight, just like the works of passion and imagination." (Dan Barbilian)

"The Mathematics bring into play spiritual powers which are not much different from those required by poetry and art." (Dan Barbilian)

Grigore C Moisil - Collected Quotes

“Throwing a small stone may have some influence on the movement of the sun"(Grigore C Moisil, “Determinism si inlantuire”, 1940)

“[…] mathematics is a science whose concepts are too breakable, too dry, too precisely limited. The disciplines of life and society, of human thinking, are fluid disciplines, with some flexibility, with concepts that are not clearly defined, but which are able to include things less strictly delimited than a mathematical definition does it.” (Grigore C Moisil, 1968) 

"All that is correct thinking is either Mathematics or likely to be reduced to Mathematics." (Grigore C Moisil)

"A mathematician does Mathematics because he sees in it something beautiful, something interesting, something he likes, something to be fond of, something that affects him, something that makes him think, meditate, dream." (Grigore C Moisil)

"A theorem is a love letter to an unknown person, to that person who catches not only its explicit meaning, but all the implicit meanings." (Grigore C Moisil)

"Learning Mathematics, one learns to think." (Grigore C Moisil)

“Logic today is not only an opportunity for philosophy, but an important instrument which people must learn to use.” (Grigore C Moisil)

"No problem has borders. Any answer has many borders." (Grigore C Moisil)

"Science is formed only from affirmations and negations, though the experiencing of science is formed from questions and answers, from hunches and doubts." (Grigore C Moisil)

"The Mathematics will be the Latin language of the future, compulsory for all scientists. Just because the Mathematics allows maximum acceleration of the movement of the scientific ideas." (Grigore C Moisil)

“The spirit of modern mathematics is based on mathematical logic, mathematical linguistics, the study of formal systems and on abstract algebra.” (Grigore C Moisil)

Romanians on Mathematics

"All that is correct thinking is either Mathematics or likely to be reduced to Mathematics." (Grigore Moisil)

"Geometry is the science which restores the situation that existed before the creation of the world and tries to fill the 'gap', relinquishing the help of matter." (Lucian Blaga)

"Learning Mathematics, one learns to think." (Grigore Moisil)

"Equality exists only in Mathematics." (Mihai Eminescu)

"Mathematics is a way of expressing the natural laws, it is the easiest and best way to describe a general law or the flow of a phenomenon, it is the most perfect language in which one can narrate a natural phenomenon." (Gheorghe Ţiţeica) 

"The mathematical works thrall and delight, just like the works of passion and imagination." (Dan Barbilian)

"The mathematician is the tamer who tamed infinity." (Lucian Blaga)

"The Mathematics bring into play spiritual powers which are not much different from those required by poetry and art." (Dan Barbilian)

"The Mathematics will be the Latin language of the future, compulsory for all scientists. Just because the Mathematics allows maximum acceleration of the movement of the scientific ideas." (Grigore Moisil)

"There is an intimate and powerful conviction among mathematicians, that supports them in their abstract researches, namely that none of their problems cannot remain without an answer." (Gheorghe Ţiţeica)

Discovery in Mathematics (unsourced)

"The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics." (Johannes Kepler)

"Indeed, when in the course of a mathematical investigation we encounter a problem or conjecture a theorem, our minds will not rest until the problem is exhaustively solved and the theorem rigorously proved; or else, until we have found the reasons which made success impossible and, hence, failure unavoidable. Thus, the proofs of the impossibility of certain solutions plays a predominant role in modern mathematics; the search for an answer to such questions has often led to the discovery of newer and more fruitful fields of endeavour." (David Hilbert)

"Mathematics originates in the mind of an individual, as it doubtless originated historically in the collective life of mankind, with the recognition of certain recurrent abstract features in common experience, and the development of processes of counting, measuring, and calculating, by which order can be brought into the manipulations of these features. It originated in this manner, indeed; but already at a very early stage it begins to transcend the practical sphere and its character undergoes a corresponding change. Intellectual curiosity progressively takes charge, despite the fact that practical considerations may for long continue to be the main source of interest and may indeed never cease to stimulate the creation of new concepts and new methods. As mathematics breaks from its early dependence on practical utility, its ‘immediate’ significance is at the same time lost and the goal is to discover what it is that makes 'emancipated' mathematics valid. (Geoffrey T Kneebone)

"No mathematician now-a-days sets any store on the discovery of isolated theorems, except as affording hints of an unsuspected new sphere of thought, like meteorites detached from some undiscovered planetary orb of speculation." (James J Sylvester)

"Those who have had the good fortune to be students of the great mathematician cannot forget the almost religious accent of his teaching, the shudder of beauty or mystery that he sent through his audience, at some admirable discovery or before the unknown." (Charles Hermite)

"The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics." (Johannes Kepler)

"Very often in mathematics the crucial problem is to recognize and discover what are the relevant concepts; once this is accomplished the job may be more than half done." (Israel N Herstein)

Discovery in Mathematics (2000-2019)

"Mathematics is about truth: discovering the truth, knowing the truth, and communicating the truth to others. It would be a great mistake to discuss mathematics without talking about its relation to the truth, for truth is the essence of mathematics. In its search for the purity of truth, mathematics has developed its own language and methodologies - its own way of paring down reality to an inner essence and capturing that essence in subtle patterns of thought. Mathematics is a way of using the mind with the goal of knowing the truth, that is, of obtaining certainty." (William Byers, "How Mathematicians Think", 2007)

"There are two aspects of proof to be borne in mind. One is that it is our lingua franca. It is the mathematical mode of discourse. It is our tried-and true methodology for recording discoveries in a bullet-proof fashion that will stand the test of time. The second, and for the working mathematician the most important, aspect of proof is that the proof of a new theorem explains why the result is true. In the end what we seek is new understanding, and ’proof’ provides us with that golden nugget." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"[…] a proof is a device of communication. The creator or discoverer of this new mathematical result wants others to believe it and accept it." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"I enjoy mathematics so much because it has a strange kind of unearthly beauty. There is a strong feeling of pleasure, hard to describe, in thinking through an elegant proof, and even greater pleasure in discovering a proof not previously known." (Martin Gardner, 2008)

"Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion - not because it makes no sense to you, but because you gave it sense and you still don't understand what your creation is up to; to have a break-through idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it." (Paul Lockhart, "A Mathematician's Lament", 2009)

"What is the basis of this interest in beauty? Is it the same in both mathematics and science? Is it rational, in either case, to expect or demand that the products of the discipline satisfy such a criterion? Is there an underlying assumption that the proper business of mathematics and science is to discover what can be discovered about reality and that truth - mathematical and physical - when seen as clearly as possible, must be beautiful? If the demand for beauty stems from some such assumption, is the assumption itself an article of blind faith? If such an assumption is not its basis, what is?" (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2010)

"[…] intuition is a very important factor in the psychology of mathematics, in the sense that mathematicians spend a great deal of time exploring guesses and checking out hunches in their efforts to discover and prove new theorems." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2010)

Discovery in Mathematics (1975-1999)

"Here is one way to look at physics: the physicists are men looking for new interpretations of the Book of Nature. After each pedestrian period of normal science, they dream up a new model, a new picture, a new vocabulary, and then they announce that the true meaning of the Book has been discovered." (Richard Rorty, "Philosophy as a Kind of Writing", 1978)

"Mathematicians do not agree among themselves whether mathematics is invented or discovered, whether such a thing as mathematical reality exists or is illusory." (Albert L Hammond, "Mathematics - Our invisible culture", 1978)

"A mathematician’s work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks." (Gian-Carlo Rota, 1981)

"For the great majority of mathematicians, mathematics is […] a whole world of invention and discovery - an art. The construction of a new theorem, the intuition of some new principle, or the creation of a new branch of mathematics is the triumph of the creative imagination of the mathematician, which can be compared to that of a poet, the painter and the sculptor." (George F J Temple, "100 Years of Mathematics: a Personal Viewpoint", 1981)

"In the initial stages of research, mathematicians do not seem to function like theorem-proving machines. Instead, they use some sort of mathematical intuition to ‘see’ the universe of mathematics and determine by a sort of empirical process what is true. This alone is not enough, of course. Once one has discovered a mathematical truth, one tries to find a proof for it." (Rudy Rucker, "Infinity and the Mind: The science and philosophy of the infinite", 1982)

"To experience the joy of mathematics is to realize mathematics is not some isolated subject that has little relationship to the things around us other than to frustrate us with unbalanced check books and complicated computations. Few grasp the true nature of mathematics - so entwined in our environment and in our lives." (Theoni Pappas, "The Joy of Mathematics" Discovering Mathematics All Around You", 1986)

"There is one qualitative aspect of reality that sticks out from all others in both profundity and mystery. It is the consistent success of mathematics as a description of the workings of reality and the ability of the human mind to discover and invent mathematical truths." (John D Barrow, "Theories of Everything: The quest for ultimate explanation. New", 1991)

"One of the lessons that the history of mathematics clearly teaches us is that the search for solutions to unsolved problems, whether solvable or unsolvable, invariably leads to important discoveries along the way." (Carl B Boyer & Uta C Merzbach, "A History of Mathematics", 1991)

"One of the deepest problems of nature is the success of mathematics as a language for describing and discovering features of physical reality." (Peter Atkins, "Creation Revisited" 1992)

"Practically everyone can understand and enjoy mathematics and appreciate its role in modem society. More generally, I feel that we develop only a small part of our potential, not only in mathematics but also in art, carpentry, cooking, drawing, singing, and so on. We close up too soon. Each of us can reach a higher level than we imagine if we are willing to explore the world and ourselves." (Sherman K Stein, "Strength in Numbers: Discovering the Joy and Power of Mathematics in Everyday Life", 1996)

"The controversy between those who think mathematics is discovered and those who think it is invented may run and run, like many perennial problems of philosophy. Controversies such as those between idealists and realists, and between dogmatists and sceptics, have already lasted more than two and a half thousand years. I do not expect to be able to convert those committed to the discovery view of mathematics to the inventionist view." (Paul Ernst, "Is Mathematics Discovered or Invented", 1996)

"There is no end to discoveries in mathematics just as there is no end to the mystery of the universe. Both are boundless. Hence mathematics is not so much a body of knowledge as a way of thought with inexhaustible possibilities." (Karma V Mital, "Understanding Mathematics And Computers", 1997)

"Mathematics is a product - a discovery - of the human mind. It enables us to see the incredible, simple, elegant, beautiful, ordered structure that lies beneath the universe we live in. It is one of the greatest creations of mankind - if it is not indeed the greatest." (Keith Devlin, "Life By the Numbers", 1998)

Discovery in Mathematics (1950-1974)

"We are driven to conclude that science, like mathematics, is a system of axioms, assumptions, and deductions; it may start from being, but later leaves it to itself, and ends in the formation of a hypothetical reality that has nothing to do with existence; or it is the discovery of an ideal being which is, of course, present in what we call actuality, and renders it an existence for us only by being present in it." (Poolla T Raju, "Idealistic Thought of India", 1953)

"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference." (George Pólya, "Induction and Analogy in Mathematics", 1954)

"At bottom, the society of scientists is more important than their discoveries. What science has to teach us here is not its techniques but its spirit: the irresistible need to explore." (Jacob Bronowski, "Science and Human Values", 1956)

"The progress of science is the discovery at each step of a new order which gives unity to what had seemed unlike." (Jacob Bronowski, "Science and Human Values", 1956)

"Is it possible to breach this wall, to present mathematics in such a way that the spectator may enjoy it? Cannot the enjoyment of mathematics be extended beyond the small circle of those who are ‘mathematically gifted’? Indeed, only a few are mathematically gifted in the sense that they are endowed with the talent to discover new mathematical facts. But by the same token, only very few are musically gifted in that they are able to compose music. Nevertheless, there are many who can understand and perhaps reproduce music, or who at least enjoy it. We believe that the number of people who can understand simple mathematical ideas is not relatively smaller than the number of those who are commonly called musical, and that their interest will be stimulated if only we can eliminate the aversion toward mathematics that so many have acquired from childhood experiences." (Hans Rademacher & Otto Toeplitz, "The Enjoyment of Mathematics", 1957)

"The heart of all major discoveries in the physical sciences is the discovery of novel methods of representation and so of fresh techniques by which inferences can be drawn - and drawn in ways which fit the phenomena under investigation." (Stephen Toulmin, "The Philosophy of Science", 1957)

“Many people think of mathematics itself as a static art - a body of eternal truth that was discovered by a few ancient, shadowy figures, and upon which engineers and scientists can draw as needed.” (Paul Halmos, “Innovation in Mathematics”, Scientific American Vol. 199 (3) , 1958) 

"There is beauty in discovery. There is mathematics in music, a kinship of science and poetry in the description of nature, and exquisite form in a molecule. Attempts to place different disciplines in different camps are revealed as artificial in the face of the unity of knowledge. All illiterate men are sustained by the philosopher, the historian, the political analyst, the economist, the scientist, the poet, the artisan, and the musician." (Glenn T Seaborg, 1958)

"The enormous usefulness of mathematics in natural sciences is something bordering on the mysterious, and there is no rational explanation for it. It is not at all natural that ‘laws of nature’ exist, much less that man is able to discover them. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." (Eugene P Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," 1960)

"No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty. This then if it has affected teaching matter, is the didactical inversion, which as it happens may be anti-didactical." (Hans Freudenthal, "The Concept and the Role of the Model in Mathematics and Natural and Social Sciences", 1961)

"It is impossible to overstate the importance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps. […] Every new discovery in mathematics, results from an attempt to solve some problem."   (Howard W Eves, "A Survey of Geometry", 1963)

"Mathematics is a creation of the mind. To begin with, there is a collection of things, which exist only in the mind, assumed to be distinguishable from one another; and there is a collection of statements about these things, which are taken for granted. Starting with the assumed statements concerning these invented or imagined things, the mathematician discovers other statements, called theorems, and proves them as necessary consequences. This, in brief, is the pattern of mathematics. The mathematician is an artist whose medium is the mind and whose creations are ideas." (Hubert S Wall, "Creative Mathematics", 1963)

"The introduction and gradual acceptance of concepts that have no immediate counterparts in the real world certainly forced the recognition that mathematics is a human, somewhat arbitrary creation, rather than an idealization of the realities in nature, derived solely from nature. But accompanying this recognition and indeed propelling its acceptance was a more profound discovery - mathematics is not a body of truths about nature." (Morris Kline, "Mathematical Thought from Ancient to Modern Times" Vol. III, 1972)

"Discovery is a double relation of analysis and synthesis together. As an analysis, it probes for what is there; but then, as a synthesis, it puts the parts together in a form by which the creative mind transcends the bare limits, the bare skeleton, that nature provides."(Jacob Bronowski, "The Ascent of Man", 1973)