Jacques S Hadamard - Collected Quotes

"Bernhard Riemann, whose extraordinary intuitive powers we have already mentioned, has especially renovated our knowledge of the distribution of prime numbers, also one of the most mysterious questions in mathematics. He has taught us to deduce results in that line from considerations borrowed from the integral calculus: more precisely, from the study of a certain quantity, a function of a variable s which may assume not only real, but also imaginary values. He proved some important properties of that function, but enunciated two or three as important ones without giving the proof. At the death of Riemann, a note was found among his papers, saying 'These properties of ζ(s) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it.' We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one. The question concerning that last one remains unsolved as yet, though, by an immense labor pursued throughout this last half century, some highly interesting discoveries in that direction have been achieved. It seems more and more probable, but still not at all certain, that the 'Riemann hypothesis' is true." (Jacques Hadamard, "The Psychology of Invention in the Mathematical Field", 1945) 

"It is important for him who wants to discover not to confine himself to one chapter of science, but to keep in touch with various others." (Jacques S Hadamard, "An Essay on the Psychology of Invention in the Mathematical Field", 1945)

"Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle." (Jacques S Hadamard, "An Essay on the Psychology of Invention in the Mathematical Field", 1945)

"The rules of algebra show that the square of any number, whether positive or negative, is a positive number: therefore, to speak of the square root of a negative number is mere absurdity. Now, Cardan deliberately commits that absurdity and begins to calculate on such 'imaginary' quantities. [...] One would describe this as pure madness; and yet the whole development of algebra and analysis would have been impossible without that fundament - which, of course, was, in the nineteenth century, established on solid and rigorous bases. It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one." (Jacque S Hadamard, "An Essay on the Psychology of Invention in the Mathematical Field", 1945)

"When I undertake some geometrical research, I have generally a mental view of the diagram itself, though generally an inadequate or incomplete one, in spite of which it affords the necessary synthesis - a tendency which, it would appear, results from a training which goes back to my very earliest childhood." (Jacques S Hadamard, "The Psychology of Invention in the Mathematical Field", 1945)

"The creation of a word or a notation for a class of ideas may be, and often is, a scientific fact of very great importance, because it means connecting these ideas together in our subsequent thought" (Jacques S Hadamard, "Newton and the Infinitesimal Calculus", 1947)

"Develop a honeybee mind, gathering ideas everywhere and associating them fully." (Jacques S Hadamard, "But You Don't Understand the Problem", Electronic News, 1967)

"Logic merely sanctions the conquests of the intuition." (Jacques S Hadamard)

"The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there never was any other object for it." (Jacques S Hadamard)

Raymond L Wilder - Collected Quotes

“In short, mathematics is what we make it; not by each of us acting without due regard for what constitutes mathematics in our culture, but by seeking to build up new theories in the light of the old, and to solve outstanding problems generally recognized as valuable for the progress of mathematics as we know it. Until we make it, it fails to ‘exist’. And, having been made, it may at some future time even fail to be ‘mathematics’ any longer.” (Raymond L Wilder, “Introduction to the Foundations of Mathematics”, 1952)

“Mathematics does not grow because a Newton, a Riemann, or a Gauss happened to be born at a certain time; great mathematicians appeared because the cultural conditions - and this includes the mathematical materials - were conducive to developing them.” (Raymond L Wilder, “Introduction to the Foundations of Mathematics”, 1952)

“The principal mathematical element in the culture, embodying the living and growing mass of modern mathematics, will be chiefly possessed by the professional mathematicians. True, certain professions, such as engineering, physics, and chemistry, which employ a great deal of mathematics, carry a sizable amount of the mathematical tradition, and in some of these, as in the case of physics and engineering research, some individuals contribute to the growth of the mathematical element in the culture. But, in the main, the mathematical element of our culture is dependent for its existence and growth on the class of those individuals known as ‘mathematicians’.” (Raymond L Wilder, “Introduction to the Foundations of Mathematics”, 1952)

“The trouble seems to lie chiefly in the assumption that mathematics is by nature something absolute, unchanging with time and place, and therefore capable of being identified once the genius with the eye sharp enough to perceive and characterize it appears on the human scene. And, since mathematics is nothing of the sort (although the layman will probably go on for centuries hence believing that it is), only failure can ensue from the attempt so to characterize it.” (Raymond L Wilder, “Introduction to the Foundations of Mathematics”, 1952)

“There is nothing mysterious, as some have tried to maintain, about the applicability of mathematics. What we get by abstraction from something can be returned.” (Raymond L Wilder, “Introduction to the Foundations of Mathematics”, 1952)

“The only reality mathematical concepts have is as cultural elements or artifacts.” (Raymond L Wilder, “Evolution of Mathematical Concepts. An Elementary Study”, 1968)

“Mathematics was born and nurtured in a cultural environment. Without the perspective which the cultural background affords, a proper appreciation of the content and state of present-day mathematics is hardly possible.” (Raymond L Wilder, American Mathematical Monthly, 1994)

Louis J Mordell - Collected Quotes

“The theory of numbers is unrivalled for the number and variety of its results and for the beauty and wealth of its demonstrations. The Higher Arithmetic seems to include most of the romance of mathematics.” (Louis J Mordell, 1917)

“Mathematical study and research are very suggestive of mountaineering. Whymper made seven efforts before he climbed the Matterhorn in the 1860’s and even then it cost the life of four of his party. Now, however, any tourist can be hauled up for a small cost, and perhaps does not appreciate the difficulty of the original ascent. So in mathematics, it may be found hard to realise the great initial difficulty of making a little step which now seems so natural and obvious, and it may not be surprising if such a step has been found and lost again.” (Louis J Mordell, “Three Lectures on Fermat’s Last Theorem”, 1921)

“What is mathematics? It has so many different aspects that the difficulties in trying to give a definition are similar to those encountered in trying to determine whether some living organisms are animal or vegetable.” (Louis J Mordell, “Reflections of a Mathematician”, 1959)

“No one will get very far or become a real mathematician without certain indispensable qualities. He must have hope, faith, and curiosity, and prime necessity is curiosity.” (Louis J Mordell, “Reflections of a Mathematician”, 1959)

“Neither you nor I nor anybody else knows what makes a mathematician tick. It is not a question of cleverness. I know many mathematicians who are far abler than I am, but they have not been so lucky. An illustration may be given by considering two miners. One may be an expert geologist, but he does not find the golden nuggets that the ignorant miner does.” (Louis J Mordell [quoted by Howard Eves, “Mathematical Circles Adieu”, 1977])

Alan M Turing - Collected Quotes

"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning. [...] The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings." (Alan M Turing, "Systems of Logic Based on Ordinals", Proceedings of the London Mathematical Society Vol 45 (2), 1939)

"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgments which are not the result of conscious trains of reasoning. These judgments are often but by no means invariably correct (leaving aside the question as to what is meant by 'correct'). Often it is possible to find some other way of verifying the correctness of an intuitive judgment. One may for instance judge that all positive integers are uniquely factorable into primes; a detailed mathematical argument leads to the same result. It will also involve intuitive judgments, but they will be ones less open to criticism than the original judgment about factorization." (Alan M Turing, "Systems of Logic Based on Ordinals", Proceedings of the London Mathematical Society Vol 45 (2), 1939)

"The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings. It is intended that when these are really well arranged the validity of the intuitive steps which are required cannot seriously be doubted." (Alan M Turing, "Systems of Logic Based on Ordinals", Proceedings of the London Mathematical Society Vol 45 (2), 1939)

"A computer would deserve to be called intelligent if it could deceive a human into believing that it was human." (Alan M Turing, "Computing Machinery and Intelligence", Mind Vol. 59, 1950)

"If one wants to make a machine mimic the behaviour of the human computer in some complex operation one has to ask him how it is done, and then translate the answer into the form of an instruction table. Constructing instruction tables is usually described as 'programming'." (Alan Turing, "Computing Machinery and Intelligence", Mind Vol. 59, 1950)

"It is unnecessary to design various new machines to do various computing processes. They can all be done with one digital computer, suitably programmed for each case." (Alan Turing, "Computing Machinery and Intelligence", Mind Vol. 59, 1950)

"The idea behind digital computers may be explained by saying that these machines are intended to carry out any operations which could be done by a human computer." (Alan M Turing, "Computing Machinery and Intelligence", Mind Vol. 59, 1950)

"The view that machines cannot give rise to surprises is due, I believe, to a fallacy to which philosophers and mathematicians are particularly subject. This is the assumption that as soon as a fact is presented to a mind all consequences of that fact spring into the mind simultaneously with it. It is a very useful assumption under many circumstances, but one too easily forgets that it is false. A natural consequence of doing so is that one then assumes that there is no virtue in the mere working out of consequences from data and general principles." (Alan M Turing, "Computing Machinery and Intelligence", Mind Vol. 59, 1950)

"This model will be a simplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge." (Alan M Turing, "The Chemical Basis of Morphogenesis" , Philosophical Transactions of the Royal Society of London, Series B: Biological Sciences, Vol. 237 (641), 1952) 

"Almost everyone now acknowledges that theory and experiment, model making, theory construction and linguistics all go together, and that the successful development of a science of behavior depends upon a ‘total approach’ in which, given that the computer ‘is the only large-scale universal model’ that we possess, ‘we may expect to follow the prescription of Simon and construct our models - or most of them - in the form of computer programs’." (Alan M Turing)

"Science is a differential equation. Religion is a boundary condition." (Alan M Turing)

"The whole thinking process is rather mysterious to us, but I believe that the attempt to make a thinking machine will help us greatly in finding out how we think ourselves." (Alan M Turing)

"We do not need to have an infinity of different machines doing different jobs. A single one will suffice. The engineering problem of producing various machines for various jobs is replaced by the office work of "programming" the universal machine to do these jobs." (Alan M Turing)

Leonardo da Vinci - Collected Quotes

"[…] no human inquiry can be called science unless it pursues its path through mathematical exposition and demonstration." (Leonardo da Vinci)

"All our knowledge has its origins in our perceptions." (Leonardo da Vinci)

"Although nature commences with reason and ends in experience it is necessary for us to do the opposite, that is to commence with experience and from this to proceed to investigate the reason." (Leonardo da Vinci)

"Experience does not feed investigators on dreams, but always proceeds from accurately determined first principles, step by step in true sequence to the end." (Leonardo da Vinci)

"He who loves practice without theory is like the sailor who boards ship without a rudder and compass and never knows where he may cast." (Leonardo da Vinci)

"Inequality is the cause of all local movements." (Leonardo da Vinci)

"Mechanics is the paradise of mathematical science, because by means of it one comes to the fruits of mathematics." (Leonardo da Vinci)

"Nature is economical and her economy is quantitative." (Leonardo da Vinci)

"Science is the observation of things possible." (Leonardo da Vinci)

"Simplicity is the ultimate sophistication." (Leonardo da Vinci)

"The grandest pleasure is the joy of understanding." (Leonardo da Vinci)

"There is no certainty where one can neither apply any of the mathematical sciences nor any of those which are connected with the mathematical sciences." (Leonardo da Vinci)

"There is no higher or lower knowledge, but one only, flowing out of experimentation." (Leonardo da Vinci)

"Those who condemn the supreme certainty of mathematics feed on confusion, and can never silence the contradictions of the sophistical sciences which lead to eternal quackery." (Leonardo da Vinci)

"To me it seems that all sciences are vain and full of errors that are not born of Experience, mother of all certainty […] that is to say, that do not at their origin, middle or end, pass through any of the five senses." (Leonardo da Vinci)

Pierre L Maupertuis - Collected Quotes

"Nature always uses the simplest means to accomplish its effects." (Pierre L Maupertuis, "Accord between different laws of Nature that seemed incompatible", Mémoires de l'académie royale des sciences, 1744)

"A true philosopher does not engage in vain disputes about the nature of motion; rather, he wishes to know the laws by which it is distributed, conserved or destroyed, knowing that such laws is the basis for all natural philosophy." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

"Despite the disorder observed in Nature, one finds enough traces of the wisdom and power of its Author that one cannot fail to recognize Him." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

"Everything is so arranged that the blind logic of mathematics executes the will of the most enlightened and free Mind." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

"One should not be deceived by philosophical works that pretend to be mathematical, but are merely dubious and murky metaphysics. Just because a philosopher can recite the words lemma, theorem and corollary doesn't mean that his work has the certainty of mathematics. That certainty does not derive from big words, or even from the method used by geometers, but rather from the utter simplicity of the objects considered by mathematics." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

"The laws of movement and of rest deduced from this principle being precisely the same as those observed in nature, we can admire the application of it to all phenomena. The movement of animals, the vegetative growth of plants [...] are only its consequences; and the spectacle of the universe becomes so much the grander, so much more beautiful, the worthier of its Author, when one knows that a small number of laws, most wisely established, suffice for all movements." (Pierre L M Maupertuis, "Les Loix du mouvement et du repos déduites d'un principe metaphysique", Histoire de l'Académie Royale des Sciences et des Belles Lettres, 1746)

"The supreme Being is everywhere; but He is not equally visible everywhere. Let us seek Him in the simplest things, in the most fundamental laws of Nature, in the universal rules by which movement is conserved, distributed or destroyed; and let us not seek Him in phenomena that are merely complex consequences of these laws." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746) 

"When a change occurs in Nature, the quantity of action necessary for that change is as small as possible." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

"To endeavor at discovering the connections that subsist in nature, is no way inconsistent with prudence; but it is downright folly to push these researches too far; as it is the lot only of superior Beings to see the dependence of events, from one end to the other, of the chain which supports them." (Pierre Louis Maupertuis, "An Essay Towards a History of the Principal Comets Since 1742", 1769)

Raymond S Nickerson - Collected Quotes

"A proof in mathematics is a compelling argument that a proposition holds without exception; a disproof requires only the demonstration of an exception. A mathematical proof does not, in general, establish the empirical truth of whatever is proved. What it establishes is that whatever is proved - usually a theorem - follows logically from the givens, or axioms." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures and Proofs", 2009)

"As well as regarding mathematics as the study of patterns, mathematics can be viewed, pragmatically, as a vast collection of problems of certain types and of approaches that have proved to be effective in solving them." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

"How are we to explain the contrast between the matter-of-fact way in which √-1 and other imaginary numbers are accepted today and the great difficulty they posed for learned mathematicians when they first appeared on the scene? One possibility is that mathematical intuitions have evolved over the centuries and people are generally more willing to see mathematics as a matter of manipulating symbols according to rules and are less insistent on interpreting all symbols as representative of one or another aspect of physical reality. Another, less self-congratulatory possibility is that most of us are content to follow the computational rules we are taught and do not give a lot of thought to rationales." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

"Philosophers have sometimes made a distinction between analytic and synthetic truths. Analytic truths are not verified by observation; true analytic statements are tautologies and are true by virtue of the definitions of their terms and their logical structure. Synthetic truths relate to the material world; the truth of synthetic statements depends on their correspondence to how physical reality works. Mathematics, according to this distinction, deals exclusively with analytic truths. Its statements are all tautologies and are (analytically) true by virtue of their adherence to formal rules of construction." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

"The characterization of mathematics as a deductive discipline is accurate but incomplete. It represents the finished and polished consequences of the work of mathematicians, but it does not adequately represent the doing of mathematics. It describes theorem proofs but not theorem proving. Moreover, the history of mathematics is not the emotionless chronology of inventions of evermore esoteric formalisms that some people imagine it to be. It has its full share of color, mystery, and intrigue." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

"The reasoning of the mathematician and that of the scientist are similar to a point. Both make conjectures often prompted by particular observations. Both advance tentative generalizations and look for supporting evidence of their validity. Both consider specific implications of their generalizations and put those implications to the test. Both attempt to understand their generalizations in the sense of finding explanations for them in terms of concepts with which they are already familiar. Both notice fragmentary regularities and - through a process that may include false starts and blind alleys - attempt to put the scattered details together into what appears to be a meaningful whole. At some point, however, the mathematician’s quest and that of the scientist diverge. For scientists, observation is the highest authority, whereas what mathematicians seek ultimately for their conjectures is deductive proof." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures and Proofs", 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", 2009)

"Without denying the usefulness of the distinction between intuition and proof, I believe it can be drawn too sharply; intuition plays an essential role in the making and evaluating of proofs and is sometimes changed as a consequence of these processes. In this respect, the distinction is like that between creative and critical thinking; while this distinction too is a useful one, it is not possible to have either in any very satisfactory sense without the other." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

Michio Kaku - Collected Quotes

"Nature is like a work by Bach or Beethoven, often starting with a central theme and making countless variations on it that are scattered throughout the symphony. By this criterion, it appears that strings are not fundamental concepts in nature." (Michio Kaku, "Hyperspace", 1995)

"No other theory known to science [other than superstring theory] uses such powerful mathematics at such a fundamental level. […] because any unified field theory first must absorb the Riemannian geometry of Einstein’s theory and the Lie groups coming from quantum field theory. […] The new mathematics, which is responsible for the merger of these two theories, is topology, and it is responsible for accomplishing the seemingly impossible task of abolishing the infinities of a quantum theory of gravity." (Michio Kaku, "Hyperspace", 1995)

"Reality has always proved to be much more sophisticated and subtle than any preconceived philosophy." (Michio Kaku, "Hyperspace", 1995)

"Remarkably, only a handful of fundamental physical principles are sufficient to summarize most of modern physics." (Michio Kaku, "Hyperspace", 1995)

"Riemann concluded that electricity, magnetism, and gravity are caused by the crumpling of our three-dimensional universe in the unseen fourth dimension. Thus a 'force' has no independent life of its own; it is only the apparent effect caused by the distortion of geometry. By introducing the fourth spatial dimension, Riemann accidentally stumbled on what would become one of the dominant themes in modern theoretical physics, that the laws of nature appear simple when expressed in higher-dimensional space. He then set about developing a mathematical language in which this idea could be expressed." (Michio Kaku, "Hyperspace", 1995)

"Scientific revolutions, almost by definition, defy common sense." (Michio Kaku, "Hyperspace", 1995)

"When Physicists speak of 'beauty' in their theories, they really mean that their theory possesses at least two essential features: 1. A unifying symmetry 2. The ability to explain vast amounts of experimental data with the most economical mathematical expressions." (Michio Kaku, "Hyperspace", 1995)

"[…] the laws of physics, carefully constructed after thousands of years of experimentation, are nothing but the laws of harmony one can write down for strings and membranes." (Michio Kaku, "Parallel Worlds: A journey through creation, higher dimensions, and the future of the cosmos", 2004)

"Chaos theory, for example, uses the metaphor of the ‘butterfly effect’. At critical times in the formation of Earth’s weather, even the fluttering of the wings of a butterfly sends ripples that can tip the balance of forces and set off a powerful storm. Even the smallest inanimate objects sent back into the past will inevitably change the past in unpredictable ways, resulting in a time paradox." (Michio Kaku, "Parallel Worlds: A journey through creation, higher dimensions, and the future of the cosmos", 2004)

"The universe is a symphony of strings, and the mind of God that Einstein eloquently wrote about for thirty years would be cosmic music resonating through eleven-dimensional hyper space." (Michio Kaku, "Parallel Worlds: A journey through creation, higher dimensions, and the future of the cosmos", 2004)

"To understand the precise point when the possible becomes the impossible, you have to appreciate and understand the laws of physics." (Michio Kaku, "The Future of the Mind: The Scientific Quest to Understand, Enhance, and Empower the Mind", 2014)