About Mathematicians (1700-1799)

"There is nothing more pleasant for man than the certainty of knowledge; whoever has once tasted of it is repelled by everything in which he perceives nothing but uncertainty. This is why the mathematicians who always deal with certain knowledge have been repelled by philosophy and other things, and have found nothing more pleasant than to spend their time with lines and letters." (Christian Wolff, 1741)

 

"In mathematics it [sophistry] had no place from the beginning: Mathematicians having had the wisdom to define accurately the terms they use, and to lay down, as axioms, the first principles on which their reasoning is grounded. Accordingly we find no parties among mathematicians, and hardly any disputes." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

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

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

"Mathematical studies […] when combined, as they now generally are, with a taste for physical science, enlarge infinitely our views of the wisdom and power displayed in the universe. The very intimate connexion indeed, which, since the date of the Newtonian philosophy, has existed between the different branches of mathematical and physical knowledge, renders such a character as that of a mere mathematician a very rare and scarcely possible occurrence." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792)

"Whoever limits his exertions to the gratification of others, whether by personal exhibition, as in the case of the actor and of the mimic, or by those kinds of literary composition which are calculated for no end but to please or to entertain, renders himself, in some measure, dependent on their caprices and humours. The diversity among men, in their judgments concerning the objects of taste, is incomparably greater than in their speculative conclusions; and accordingly, a mathematician will publish to the world a geometrical demonstration, or a philosopher, a process of abstract reasoning, with a confidence very different from what a poet would feel, in communicating one of his productions even to a friend." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792)

"So-called professional mathematicians have, in their reliance on the relative incapacity of the rest of mankind, acquired for themselves a reputation for profundity very similar to the reputation for sanctity possessed by theologians." (Georg C Lichtenberg, "Aphorisms", 1765-1799)

Mathematicians vs. Physicists II

"The domain of physics is no proper field for mathematical pastimes. The best security would be in giving a geometrical training to physicists, who need not then have recourse to mathematicians, whose tendency is to despise experimental science." (Auguste Comte, "The Positive Philosophy", 1830)

"So intimate is the union between Mathematics and Physics that probably by far the larger part of the accessions to our mathematical knowledge have been obtained by the efforts of mathematicians to solve the problems set to them by experiment, and to create for each successive class phenomena a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature. Sometimes the mathematician has been before the physicist, and it has happened that when some great and new question has occurred to the experimentalist or the observer, he has found in the armory of the mathematician the weapons which he needed ready made to his hand. But much oftener, the questions proposed by the physicist have transcended the utmost powers of the mathematics of the time, and a fresh mathematical creation has been needed to supply the logical instrument requisite to interpret the new enigma." (Henry J S Smith, Nature, Volume 8, 1873) 

"At the present time it is of course quite customary for physicists to trespass on chemical ground, for mathematicians to do excellent work in physics, and for physicists to develop new mathematical procedures […] Trespassing is one of the most successful techniques in science." (Wolfgang Köhler, "Dynamics in Psychology", 1940)

"It is positively spooky how the physicist finds the mathematician has been there before him or her." (Steven Weinberg,"Lectures on the Applicability of Mathematics" , Notices of the American Mathematical Society, 1986) 

"Mathematician ought not to be for the physicist a simple provider of formulae."(Henri Poincaré, The Relations of Analysis and Mathematical Physics, Bulletin of the American Mathematical Society, Volume 4 (6), 1896)

"Probably, what characterizes all scientists, whatever they may be, archivists, mathematicians, chemists, astronomists, physicists, is that they do not seek to reach a practical conclusion by their work." (Charles Richet, "The Natural History of a Savant", 1927)

"[...] the mathematical physicist [...] obtains much prestige from the physicists because they are impressed with the amount of mathematics he knows, and much prestige from the mathematicians, because they are impressed with the amount of physics he knows." (William F G Swann, "The Architecture of the Universe", 1934) 

"Mathematicians who build new spaces and physicists who find them in the universe can profit from the study of pictorial and architectural spaces conceived and built by men of art." (György Kepes, "The New Landscape In Art and Science", 1956)

"Physicists are more like avant-garde composers, willing to bend traditional rules and brush the edge of acceptability in the search for solutions. Mathematicians are more like classical composers, typically working within a much tighter framework, reluctant to go to the next step until all previous ones have been established with due rigor. Each approach has its advantages as well as drawbacks; each provides a unique outlet for creative discovery. Like modern and classical music, it’s not that one approach is right and the other wrong – the methods one chooses to use are largely a matter of taste and training." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 1999)

"Theoretical physicists are like pure mathematicians, in that they are often interested in the hypothetical behaviour of entirely imaginary objects, such as parallel universes, or particles traveling faster than light, whose actual existence is not being seriously proposed at all." (John Ziman," Real Science: What it Is, and what it Means", 2000) 

Mathematicians vs. Physicists I

"Mathematicians will do well to observe that a reasonable acquaintance with theoretical physics at its present stage of development, to mention only such broad subjects as electricity, elastics, hydrodynamics, etc., is as much as most of us can keep permanently assimilated. It should also be remembered that the step from the formal elegance of theory to the brute arithmetic of the special case is always humiliating, and that this labor usually falls to the lot of the physicist." (Carl Barus, "The Mathematical Theory of the Top", 1898)

"Our environment may and should mean something towards us which is not to be measured with the tools of the physicist or described by the metrical symbols of the mathematician." (Arthur S Eddington, "Science and the Unseen World", 1929)

"[…] there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, [...] the mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real', but [...] [a physicist] is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics." (Godfrey H Hardy, "A Mathematician's Apology", 1940)

"It is to be hoped that in the future more and more theoretical physicists will command a deep knowledge of mathematical principles; and also that mathematicians will no longer limit themselves so exclusively to the aesthetic development of mathematical abstractions." (George D Birkhoff, "Mathematical Nature of Physical Theories" American Scientific Vol. 31 (4), 1943)

"The mathematicians know a great deal about very little and the physicists very little about a great deal." (Stanislaw Ulam, "On the Ergodic Behavior of Dynamical Systems", 1955) 

"The mathematicians and physics men Have their mythology; they work alongside the truth, Never touching it; their equations are false But the things work. Or, when gross error appears, They invent new ones; they drop the theory of waves In universal ether and imagine curved space." (Robinson Jeffers," The Beginning and the End and Other Poems, The Great Wound", 1963) 

"When the problems in physics become difficult we may often look to the mathematician who may already have studied such things and have prepared a line of reasoning for us to follow. On the other hand they may not have, in which case we have to invent our own line of reasoning, which we then pass back to the mathematician." (Richard Feynman,"The Character of Physical Law", 1965)

"Empirical evidence can never establish mathematical existence--nor can the mathematician's demand for existence be dismissed by the physicist as useless rigor. Only a mathematical existence proof can ensure that the mathematical description of a physical phenomenon is meaningful." (Richard Courant, "The Parsimonious Universe, Stefan Hildebrandt & Anthony Tromba", 1996) 

"[…] mathematicians are much more concerned for example with the structure behind something or with the whole edifice. Mathematicians are not really puzzlers. Those who really solve mathematical puzzles are the physicists. If you like to solve mathematical puzzles, you should not study mathematics but physics!" (Carlo Beenakker, [interview] 2006)

"That is, the physicist likes to learn from particular illustrations of a general abstract concept. The mathematician, on the other hand, often eschews the particular in pursuit of the most abstract and general formulation possible. Although the mathematician may think from, or through, particular concrete examples in coming to appreciate the likely truth of very general statements, he will hide all those intuitive steps when he comes to present the conclusions of his thinking to outsiders. It presents the results of research as a hierarchy of definitions, theorems and proofs after the manner of Euclid; this minimizes unnecessary words but very effectively disguises the natural train of thought that led to the original results." (John D Barrow, "New Theories of Everything", 2007)

About Mathematicians (1920-1929)

"It is not surprising that the greatest mathematicians have again and again appealed to the arts in order to find some analogy to their own work. They have indeed found it in the most varied arts, in poetry, in painting, and in sculpture, although it would certainly seem that it is in music, the most abstract of all the arts, the art of number and of time, that we find the closest analogy." (Havelock Ellis, "The Dance of Life", 1923)

"The mathematician has reached the highest rung on the ladder of human thought." (Havelock Ellis, "The Dance of Life", 1923)

"Proof is an idol before whom the pure mathematician tortures himself." (Arthur S Eddington, 1927)

"I have myself always thought of a mathematician as in the first instance an observer, a man who gazes at a distant range of mountains and notes down his observations. His object is simply to distinguish clearly and notify to others as many different peaks as he can.” (Godfrey H Hardy, “Mathematical Proof”, Mind 38, 1929)

"Our environment may and should mean something towards us which is not to be measured with the tools of the physicist or described by the metrical symbols of the mathematician." (Arthur S Eddington, "Science and the Unseen World", 1929) 

About Mathematicians (Unsourced)

"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 mathematician, then, will be defined in what follows as someone who has published the proof of at least one non-trivial theorem." (Jean Dieudonné)

"All mathematicians live in two different worlds. They live in a crystalline world of perfect platonic forms. An ice palace. But they also live in the common world where things are transient, ambiguous, subject to vicissitudes. Mathematicians go backward and forward from one world to another. They’re adults in the crystalline world, infants in the real one." (Sylvain Cappell)

"Empirical evidence can never establish mathematical existence—nor can the mathematician's demand for existence be dismissed by the physicist as useless rigor. Only a mathematical existence proof can ensure that the mathematical description of a physical phenomenon is meaningful." (Richard Courant)

"Good mathematicians see analogies between theorems or theories, the very best ones see analogies between analogies." (Stefan Banach)

"Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness." (Eric T Bell)

"I do believe in simplicity. It is astonishing as well as sad, how many trivial affairs even the wisest thinks he must attend to in a day; how singular an affair he thinks he must omit. When the mathematician would solve a difficult problem, he first frees the equation of all encumbrances, and reduces it to its simplest terms. So simplify the problem of life, distinguish the necessary and the real. Probe the earth to see where your main roots run." (Henry David Thoreau)

"In my opinion a mathematician, in so far as he is a mathematician, need not preoccupy himself with philosophy - an opinion, moreover, which has been expressed by many philosophers." (Henri Lebesgue)

"In the broad light of day mathematicians check their equations and their proofs, leaving no stone unturned in their search for rigour. But, at night, under the full moon, they dream, they float among the stars and wonder at the miracle of the heavens. They are inspired. Without dreams there is no art, no mathematics, no life." (Michael Atiyah, "The Art of Mathematics" [in "Art in the Life of Mathematicians"])

"Mathematicians have never been in full agreement on their science, though it is said to be the science of self-evident verities - absolute, indisputable and definitive. They have always been in controversy over the developing aspects of mathematics, and they have always considered their own age to be a period of crisis." (Henri Lebesgue)

"Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only." (Henri Poincaré)

"Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty's sake and pulls it down to earth." (Marston Morse)

"Mathematics is the life supreme. The life of the gods is mathematics. All divine messengers are mathematicians. Pure mathematics is religion. Its attainment requires a theophany." (Friederich von Hardenberg [Novalis])

"More than any other science, mathematics develops through a sequence of consecutive abstractions. A desire to avoid mistakes forces mathematicians to find and isolate the essence of problems and the entities considered. Carried to an extreme, this procedure justifies the well-known joke that a mathematician is a scientist who knows neither what he is talking about nor whether whatever he is talking about exists or not." (Élie Cartan) 

"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])

"No mathematician nowadays 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)

"On foundations we believe in the reality of mathematics, but of course, when philosophers attack us with their paradoxes, we rush to hide behind formalism and say 'mathematics is just a combination of meaningless symbols’ […]. Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. The sensation is probably an illusion, but it is very convenient." (Jean Dieudonné)

"So if you could be the Devil and offer a mathematician to sell his soul for the proof of one theorem - what theorem would most mathematicians ask for?" (H Montgomery)

"The mathematical method is the essence of mathematics. He who fully comprehends the method is a mathematician." (Friederich von Hardenberg [Novalis])

"The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God." (John W N Sullivan)

"The mathematician's best work is art […] a high and perfect art, as daring as the most secret dreams of imagination, clear and limpid. Mathematical genius and artistic genius touch each other." (Gustav Mittag-Leffler)

"The mathematician’s eye is a mystic mirror, not only reflecting reality but absorbing it." (Francis O Googol)

"[…] the mathematician learns early to accept no fact, to believe no statement, however apparently reasonable or obvious or trivial, until it has been proved, rigorously and totally by a series of steps proceeding from universally accepted first principles." (Alfred Adler)

"The real mathematician is an enthusiast per se. Without enthusiasm no mathematics." (Friederich von Hardenberg [Novalis])

"The science of physics does not only give us [mathematicians] an oportunity to solve problems, but helps us also to discover the means of solving them, and it does this in two ways: it leads us to anticipate the solution and suggests suitable lines of argument." (Henri Poincaré)

"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)

"What makes a great mathematician? A feel for form, a strong sense of what is important. Möbius had both in abundance. He knew that topology was important. He knew that symmetry is a fundamental and powerful mathematical principle. The judgment of posterity is clear: Möbius was right." (Ian Stewart)

About Mathematicians (2010-2019)

"In the broad light of day mathematicians check their equations and their proofs, leaving no stone unturned in their search for rigour. But, at night, under the full moon, they dream, they float among the stars and wonder at the miracle of the heavens. They are inspired. Without dreams there is no art, no mathematics, no life." (Michael F Atiyah, "The Art of Mathematics", 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)

"A surprising proportion of mathematicians are accomplished musicians. Is it because music and mathematics share patterns that are beautiful?" (Martin Gardner, "The Dover Math and Science Newsletter", 2011)

"To get a true understanding of the work of mathematicians, and the need for proof, it is important for you to experiment with your own intuitions, to see where they lead, and then to experience the same failures and sense of accomplishment that mathematicians experienced when they obtained the correct results. Through this, it should become clear that, when doing any level of mathematics, the roads to correct solutions are rarely straight, can be quite different, and take patience and persistence to explore.” (Alan Sultan & Alice F Artzt, “The Mathematics that every Secondary School Math Teacher Needs to Know”, 2011)

“Often the key contribution of intuition is to make us aware of weak points in a problem, places where it may be vulnerable to attack. A mathematical proof is like a battle, or if you prefer a less warlike metaphor, a game of chess. Once a potential weak point has been identified, the mathematician’s technical grasp of the machinery of mathematics can be brought to bear to exploit it.” (Ian Stewart, “Visions of Infinity”, 2013)

"Proof, in fact, is the requirement that makes great problems problematic. Anyone moderately competent can carry out a few calculations, spot an apparent pattern, and distil its essence into a pithy statement. Mathematicians demand more evidence than that: they insist on a complete, logically impeccable proof. Or, if the answer turns out to be negative, a disproof. It isn’t really possible to appreciate the seductive allure of a great problem without appreciating the vital role of proof in the mathematical enterprise. Anyone can make an educated guess. What’s hard is to prove it’s right. Or wrong.” (Ian Stewart, "Visions of Infinity", 2013)

"Models can be: formulations, abstractions, replicas, idealizations, metaphors - and combinations of these. [...] Some mathematical models have been blindly used - their presuppositions as little understood as any legal fine print one ‘agrees to’ but never reads - with faith in their trustworthiness. The very arcane nature of some of the formulations of these models might have contributed to their being given so much credence. If so, we mathematicians have an important mission to perform: to help people who wish to think through the fundamental assumptions underlying models that are couched in mathematical language, making these models intelligible, rather than (merely) formidable Delphic oracles.” (Barry Mazur, "The Authority of the Incomprehensible" , 2014)

"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)

"Mathematical rigour is the thing that enables mathematicians to agree with one another about what is and isn’t correct, rather than just having arguments about competing theories and never coming to a conclusion. Mathematics is based on the rules of logic, the idea being that if you only use objects that behave strictly according to the rules of logic, then as long as you only strictly apply the rules of logic, no disagreements can ever arise."(Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Mathematicians start by playing around with ideas to get a feel for what might be possible, good and bad." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Mathematics often develops by mathematicians feeling frustrated about being unable to do something in the existing world, so they invent a new world in which they can do it."(Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"The Axiom of Choice says that it is possible to make an infinite number of arbitrary choices. […] Mathematicians don’t exactly care whether or not the Axiom of Choice holds over all, but they do care whether you have to use it in any given situation or not." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

About Mathematicians (2000-2009)

“Mathematics is not placid, static and eternal. […] Most mathematicians are happy to make use of those axioms in their proofs, although others do not, exploring instead so-called intuitionist logic or constructivist mathematics. Mathematics is not a single monolithic structure of absolute truth!” (Gregory J Chaitin, “A century of controversy over the foundations of mathematics”, 2000)

"It is sometimes said that mathematics is not an experimental subject. This is not true! Mathematicians often use the evidence of lots of examples to help form a conjecture, and this is an experimental approach. Having formed a conjecture about what might be true, the next task is to try to prove it." (George M Phillips,"Mathematics Is Not a Spectator Sport" , 2000)

"By common consensus in the mathematical world, a good proof displays three essential characteristics: a good proof is (1) convincing, (2) surveyable, and (3) formalizable. The first requirement means simply that most mathematicians believe it when they see it. […] Most mathematicians and philosophers of mathematics demand more than mere plausibility, or even belief. A proof must be able to be understood, studied, communicated, and verified by rational analysis. In short, it must be surveyable. Finally, formalizability means we can always find a suitable formal system in which an informal proof can be embedded and fleshed out into a formal proof." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"Somehow mathematicians seem to long for more than just results from their proofs; they want insight." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"The real raison d'etre for the mathematician's existence is simply to solve problems. So what mathematics really consists of is problems and solutions. And it is the 'good' problems, the ones that challenge the greatest minds for decades, if not centuries, that eventually become enshrined as mathematical mountaintops." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"[…] a mathematician is more anonymous than an artist. While we may greatly admire a mathematician who discovers a beautiful proof, the human story behind the discovery eventually fades away and it is, in the end, the mathematics itself that delights us." (Timothy Gowers, "Mathematics: A Very Short Introduction", 2002)

"To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. Mathematics is the ultimate in technology transfer." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)

"[…] all human beings - professional mathematicians included - are easily muddled when it comes to estimating the probabilities of rare events. Even figuring out the right question to ask can be confusing." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"Elegance and simplicity should remain important criteria in judging mathematics, but the applicability and consequences of a result are also important, and sometimes these criteria conflict. I believe that some fundamental theorems do not admit simple elegant treatments, and the proofs of such theorems may of necessity be long and complicated. Our standards of rigor and beauty must be sufficiently broad and realistic to allow us to accept and appreciate such results and their proofs. As mathematicians we will inevitably use such theorems when it is necessary in the practice our trade; our philosophy and aesthetics should reflect this reality." (Michael Aschbacher, "Highly complex proofs and implications", 2005)

"At some point in his or her life every working mathematician has to explain to someone, usually a relative, that mathematics is hardly a finished project. Mathematicians know, of course, that it is far too soon to put the glorious achievements of their trade into a big museum and just become happy curators. In many respects, the study of mathematics has hardly begun." (Barry Mazur, [foreword] 2006)

"[…] mathematicians are much more concerned for example with the structure behind something or with the whole edifice. Mathematicians are not really puzzlers. Those who really solve mathematical puzzles are the physicists. If you like to solve mathematical puzzles, you should not study mathematics but physics!" (Carlo Beenakker, [interview] 2006)

"Many people believe that all of mathematics has already been discovered and codified. Mathematicians (they think) do nothing except rearrange the material in different ways for different types of students. This seems to be the result of the cut-and-dried method of teaching mathematics in many high schools and universities. The facts are laid out in the cleanest logical order. Little attempt is made to show how someone once had to invent it all, at first in a confused way, and that only later was it possible to give it this neat form." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"[…] mathematicians complain when physicists leap over technicalities, such as throwing away terms they don’t like in differential equations. […] Mathematicians worry about justifying […] approximations and spend a lot of effort coping with paranoid delusions […] Mathematicians cherish the rare moments where physicists’ leaps of faith get them into trouble. […] While it is fun to point out physicists’ errors, it is much more satisfying when wediscover something that they don’t know." (Rick Durrett, "Random Graph Dynamics", 2006)

"Mathematicians often get bored by a problem after they have fully understood it and have given proofs of their conjectures. Sometimes they even forget the precise details of what they have done after the lapse of years, having refocused their interest in another area. The common notion of the mathematician contemplating timeless truths, thinking over the same proof again and again - Euclid looking on beauty bare - is rarely true in any static sense." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"One thing mathematicians do is connect concepts that occur in different trains of thought." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Still, in the end, we find ourselves drawn to the beauty of the patterns themselves, and the amazing fact that we humans are smart enough to prove even a feeble fraction of all possible theorems about them. Often, greater than the contemplation of this beauty for the active mathematician is the excitement of the chase. Trying to discover first what patterns actually do or do not occur, then finding the correct statement of a conjecture, and finally proving it - these things are exhilarating when accomplished successfully. Like all risk-takers, mathematicians labor months or years for these moments of success." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Equations are the mathematician's way of working out the value of some unknown quantity from circumstantial evidence. ‘Here are some known facts about an unknown number: deduce the number.’ An equation, then, is a kind of puzzle, centered upon a number. We are not told what this number is, but we are told something useful about it. Our task is to solve the puzzle by finding the unknown number." (Ian Stewart, “Why Beauty Is Truth”, 2007)

“Mathematicians, then, do not just care about proving theorems: they care about proving interesting, deep, fruitful theorems, by means of elegant, ingenious, explanatory, memorable, or even amusing proofs. If we wish to understand more about the character of mathematical knowledge, we ought to investigate these kinds of evaluative claims made by mathematicians.” (Mary Leng [“Mathematical Knowledge”, Ed. by Mary Leng, Alexander Paseau and Michael Potter], 2007)

"Mathematics as done by mathematicians is not just heaping up statements logically deduced from the axioms. Most such statements are rubbish, even if perfectly correct. A good mathe matician will look for interesting results. These interesting results, or theorems, organize themselves into meaningful and natural structures, and one may say that the object of mathematics is to find and study these structures." (David Ruelle, "The Mathematician's Brain", 2007)

"Mathematics is useful. It is the language of physics, and some aspects of mathematics are important in all the sciences and their applications and also in finance. But my personal experience is that good mathematicians are rarely pushed by a high sense of duty and achievement that would urge them to do something useful. In fact, some mathematicians prefer to think that their work is absolutely useless." (David Ruelle, "The Mathematician's Brain", 2007)

“Popular accounts of mathematics often stress the discipline’s obsession with certainty, with proof. And mathematicians often tell jokes poking fun at their own insistence on precision. However, the quest for precision is far more than an end in itself. Precision allows one to reason sensibly about objects outside of ordinary experience. It is a tool for exploring possibility: about what might be, as well as what is.” (Donal O’Shea, “The Poincaré Conjecture”, 2007)

“The infinite more than anything else is what characterizes mathematics and defines its essence. […] To grapple with infinity is one of the bravest and extraordinary endeavors that human beings have ever undertaken.” (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)

"That is, the physicist likes to learn from particular illustrations of a general abstract concept. The mathematician, on the other hand, often eschews the particular in pursuit of the most abstract and general formulation possible. Although the mathematician may think from, or through, particular concrete examples in coming to appreciate the likely truth of very general statements, he will hide all those intuitive steps when he comes to present the conclusions of his thinking to outsiders. It presents the results of research as a hierarchy of definitions, theorems and proofs after the manner of Euclid; this minimizes unnecessary words but very effectively disguises the natural train of thought that led to the original results." (John D Barrow, "New Theories of Everything", 2007)

"We must be prepared to find that the perfection, purity, and simplicity that we love in mathematics is metaphorically related to a yearning for human perfection, purity, and simplicity. And this may explain why mathematicians often have a religious inclination. But we must also be prepared to find that our love of mathematics is not exempt from the usual human contradictions." (David Ruelle, "The Mathematician's Brain", 2007)

“If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.” (Paul Lockhart, “A Mathematician's Lament", 2009) 

"Mathematicians seek a certain kind of beauty. Perhaps mathematical beauty is a constant - as far as the contents of mathematics are concerned - and yet the forms this beauty takes are certainly cultural. And while the history of mathematics surely is many stranded, one of its most important strands is formed by such cultural forms of mathematical beauty.” (Reviel Netz, “Ludic Proof: Greek Mathematics and the Alexandrian Aesthetic”, 2009)

“Mathematicians are sometimes described as living in an ideal world of beauty and harmony. Instead, our world is torn apart by inconsistencies, plagued by non sequiturs and, worst of all, made desolate and empty by missing links between words, and between symbols and their referents; we spend our lives patching and repairing it. Only when the last crack disappears are we rewarded by brief moments of harmony and joy.” (Alexandre V Borovik, “Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice”, 2009)

“Obviously, the final goal of scientists and mathematicians is not simply the accumulation of facts and lists of formulas, but rather they seek to understand the patterns, organizing principles, and relationships between these facts to form theorems and entirely new branches of human thought.” (Clifford A Pickover, “The Math Book”, 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)

About Mathematicians (1930-1939)

"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories." (Stefan Banach, cca. 1930)

"Between the philosopher's attitude towards the issue of reality and that of the mathematician there is this essential difference: for the philosopher the issue is paramount; the mathematician's love for reality is purely platonic." (Tobias Dantzig, "Number: The Language of Science", 1930)

"Mathematics has been called the science of the infinite. Indeed, the mathematician invents finite constructions by which questions are decided that by their very nature refer to the infinite. This is his glory." (Hermann Weyl, "Levels of Infinity", cca. 1930)

“In pure mathematics the maximum of detachment appears to be reached: the mind moves in an infinitely complicated pattern, which is absolutely free from temporal considerations. Yet this very freedom – the essential condition of the mathematician’s activity – perhaps gives him an unfair advantage. He can only be wrong – he cannot cheat.” (Kytton Strachey, “Portraits in Miniature”, 1931)

"If a mathematician wishes to disparage the work of one of his colleagues, say, A, the most effective method he finds for doing this is to ask where the results can be applied. The hard pressed man, with his back against the wall, finally unearths the researches of another mathematician B as the locus of the application of his own results. If next B is plagued with a similar question, he will refer to another mathematician C. After a few steps of this kind we find ourselves referred back to the researches of A, and in this way the chain closes." (Alfred Tarski, "The Semantic Conception of Truth", 1933)

“To create a healthy philosophy you should renounce metaphysics but be a good mathematician.” (Bertrand Russell, [lecture] 1935)

"Mathematicians and other scientists, however great they may be, do not know the future. Their genius may enable them to project their purpose ahead of them; it is as if they had a special lamp, unavailable to lesser men, illuminating their path; but even in the most favorable cases the lamp sends only a very small cone of light into the infinite darkness." (George Sarton, "The Study of the History of Mathematics", 1936)

"The concatenations of mathematical ideas are not divorced from life, far from it, but they are less influenced than other scientific ideas by accidents, and it is perhaps more possible, and more permissible, for a mathematician than for any other man to secrete himself in a tower of ivory." (George Sarton, "The Study of the History of Mathematics", 1936)

"The main source of mathematical invention seems to be within man rather than outside of him: his own inveterate and insatiable curiosity, his constant itching for intellectual adventure; and likewise the main obstacles to mathematical progress seem to be also within himself; his scandalous inertia and laziness, his fear of adventure, his need of conformity to old standards, and his obsession by mathematical ghosts." (George Sarton, "The Study of the History of Mathematics", 1936)

"Mathematicians study their problems on account of their intrinsic interest, and develop their theories on account of their beauty.” (Karl Menger, The Scientific Monthly, 1937)

"We can now return to the distinction between language and symbolism. A symbol is language and yet not language. A mathematical or logical or any other kind of symbol is invented to serve a purpose purely scientific; it is supposed to have no emotional expressiveness whatever. But when once a particular symbolism has been taken into use and mastered, it reacquires the emotional expressiveness of language proper. Every mathematician knows this. At the same time, the emotions which mathematicians find expressed in their symbols are not emotions in general, they are the peculiar emotions belonging to mathematical thinking." (Robin G Collingwood, "The Principles of Art", 1938)

"Pure mathematics and physics are becoming ever more closely connected, though their methods remain different. One may describe the situation by saying that the mathematician plays a game in which he himself invents the rules while the while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen." (Paul A M Dirac, "The Relation Between Mathematics and Physics", Proceedings of the Royal Society of Edinburgh, 1938-1939)

About Mathematicians (1990-1999)

"A mathematician, then, will be defined in what follows as someone who has published the proof of at least one non-trivial theorem." (Jean Dieudonné, "Mathematics and Mathematicians", 1992)

"Engineers, always looking for optimal values for the measures of magnitudes which interest them, think of mathematicians as custodians of a fund of formulae, to be supplied to them on demand." (Jean Dieudonné, "Mathematics - The Music of Reason", 1992)

"Every mathematician worthy of the name has experienced the state of lucid exaltation in which one thought succeeds another as if miraculously. This feeling may last for hours at a time, even for days. Once you have experienced it, you are eager to repeat it but unable to do it at will, unless perhaps by dogged work." (André Weil, "The Apprenticeship of a Mathematician", 1992)

"In the flowering of a mathematical talent social environment has an important part to play." (Jean Dieudonné, "Mathematics - The Music of Reason", 1992)

"The life of a mathematician is dominated by an insatiable curiosity, a desire bordering on passion to solve the problems he is studying." (Jean Dieudonné, "Mathematics - The Music of Reason", 1992)

"[...] there is no criterion for appreciation which does not vary from one epoch to another and from one mathematician to another. [...] These divergences in taste recall the quarrels aroused by works of art, and it is a fact that mathematicians often discuss among themselves whether a theorem is more or less ‚beautiful‘. This never fails to surprise practitioners of other sciences: for them the sole criterion is the 'truth' of a theory or formula." (Jean Dieudonné, "Mathematics - The Music of Reason", 1992)

"To a mathematician, an object possesses symmetry if it retains its form after some transformation. A circle, for example, looks the same after any rotation; so a mathematician says that a circle is symmetric, even though a circle is not really a pattern in the conventional sense - something made up from separate, identical bits. Indeed the mathematician generalizes, saying that any object that retains its form when rotated - such as a cylinder, a cone, or a pot thrown on a potter's wheel - has circular symmetry." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"Virtually all mathematical theorems are assertions about the existence or nonexistence of certain entities. For example, theorems assert the existence of a solution to a differential equation, a root of a polynomial, or the nonexistence of an algorithm for the Halting Problem. A platonist is one who believes that these objects enjoy a real existence in some mystical realm beyond space and time. To such a person, a mathematician is like an explorer who discovers already existing things. On the other hand, a formalist is one who feels we construct these objects by our rules of logical inference, and that until we actually produce a chain of reasoning leading to one of these objects they have no meaningful existence, at all." (John L Casti, "Reality Rules: Picturing the world in mathematics" Vol. II, 1992)

"Mathematicians tells us that it is easy to invent mathematical theorems which are true, but that it is hard to find interesting ones. In analyzing music or writing its history, we meet the same difficulty, and it is compounded by another." (Charles Rosen, "The Frontiers of Meaning: Three Informal Lectures on Music", 1994)

"The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It's like building a bridge. Once the main lines of the structure are right, then the details miraculously fit. The problem is the overall design." (Freeman J Dyson, [interview] 1994)

"Ironically, mathematicians often infer all sorts of properties about objects which they only suspect, or hope, actually exist. If their suspicions turn out to be unfounded, then they seem to end up by knowing rather a lot about something which does not exist and which might seem, therefore, not to have any properties at all." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mathematicians get a different kind of pleasure from the illumination of solving a problem, when what was once mysterious and obscure is made plain. Revealing the hidden connections in a situation is delightful - like reaching the top of a mountain after a hard climb, and seeing the landscape spread out before you. All of a sudden, everything is clear! If the result is extremely simple, so much the better . To start with confusing complexity and transform it into revealing simplicity is a marvellous reward for hard work. It really does give the mathematician a 'kick'!" (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mystery is found as much in mathematics as in detective stories. Indeed, the mathematician could well be described as a detective, brilliantly exploiting a few initial clues to solve the problem and reveal its innermost secrets. An especially mathematical mystery is that you can often search for some mathematical object, and actually know a lot about it, if it exists, only to discover that in fact it does not exist at all - you knew a lot about something which cannot be." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"One of the greatest delights of mathematics is finding unity among apparent diversity; realizing that situations (or objects) that seemed to be quite different are actually basically the same. Spotting unexpected connections is not only a source of pleasure, it is an essential step in the development of mathematics. Without making connections, mathematics would quickly turn into a collection of separate topics, studied by mathematicians who had nothing to say to colleagues outside their own specialty." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"The entrepreneur's instinct is to exploit the natural world. The engineer's instinct is to change it. The scientist's instinct is to try to understand it - to work out what's really going on. The mathematician's instinct is to structure that process of understanding by seeking generalities that cut across the obvious subdivisions." (Ian Stewart, "Nature's Numbers", 1995)

"Empirical evidence can never establish mathematical existence--nor can the mathematician's demand for existence be dismissed by the physicist as useless rigor. Only a mathematical existence proof can ensure that the mathematical description of a physical phenomenon is meaningful." (Richard Courant, "The Parsimonious Universe, Stefan Hildebrandt & Anthony Tromba", 1996)

"The whole thing that makes a mathematician's life worthwhile is that he gets the grudging admiration of three or four colleagues." (Donald E Knuth, [interview] 1996)

"To be an engineer, and build a marvelous machine, and to see the beauty of its operation is as valid an experience of beauty as a mathematician's absorption in a wondrous theorem. One is not ‘more’ beautiful than the other. To see a space shuttle standing on the launch pad, the vented gases escaping, and witness the thunderous blast-off as it climbs heavenward on a pillar of flame - this is beauty. Yet it is a prime example of applied mathematics.” (Calvin C Clawson, “Mathematical Mysteries”, 1996)

“In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religions and mathematics attempt to express the relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, and impenetrable language. Both exercise the deep recesses of our mind and stimulate our imagination. Mathematicians, like priests, seek ‘ideal’, immutable, nonmaterial truths and then often try to apply theses truth in the real world.” (Clifford A Pickover, "The Loom of God: Mathematical Tapestries at the Edge of Time", 1997)

“The most common instance of beauty in mathematics is a brilliant step in an otherwise undistinguished proof. […] A beautiful theorem may not be blessed with an equally beautiful proof; beautiful theorems with ugly proofs frequently occur. When a beautiful theorem is missing a beautiful proof, attempts are made by mathematicians to provide new proofs that will match the beauty of the theorem, with varying success. It is, however, impossible to find beautiful proofs of theorems that are not beautiful.” (Gian-Carlo Rota, “The Phenomenology of Mathematical Beauty”, 1997)

"Everybody knows that mathematics is about Miracles, only mathematicians have a name for them: Theorems." (Roger Howe, 1998)

"Rather mathematicians like to look for patterns, and the primes probably offer the ultimate challenge. When you look at a list of them stretching off to infinity, they look chaotic, like weeds growing through an expanse of grass representing all numbers. For centuries mathematicians have striven to find rhyme and reason amongst this jumble. Is there any music that we can hear in this random noise? Is there a fast way to spot that a particular number is prime? Once you have one prime, how much further must you count before you find the next one on the list? These are the sort of questions that have tantalized generations." (Marcus du Sautoy, "The Music of the Primes", 1998)

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

"Mathematicians, like the rest of us, cherish clever ideas; in particular they delight in an ingenious picture. But this appreciation does not overwhelm a prevailing skepticism. After all, a diagram is - at best - just a special case and so can't establish a general theorem. Even worse, it can be downright misleading. Though not universal, the prevailing attitude is that pictures are really no more than heuristic devices; they are psychologically suggestive and pedagogically important - but they prove nothing. I want to oppose this view and to make a case for pictures having a legitimate role to play as evidence and justification - a role well beyond the heuristic.  In short, pictures can prove theorems." (James R Brown, "Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures", 1999)

"Physicists are more like avant-garde composers, willing to bend traditional rules and brush the edge of acceptability in the search for solutions. Mathematicians are more like classical composers, typically working within a much tighter framework, reluctant to go to the next step until all previous ones have been established with due rigor. Each approach has its advantages as well as drawbacks; each provides a unique outlet for creative discovery. Like modern and classical music, it’s not that one approach is right and the other wrong – the methods one chooses to use are largely a matter of taste and training." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 1999)

“The spirit of mathematics and the essence of its beauty is remarkably fragile, because mathematics is about ideas and about thought. Mathematics takes place in the mind, and no two minds are the same. After many years of study and work, a mathematician may stumble on a vast and beautiful vista that unifies and simplifies many things that once appeared disparate and complicated. Mathematicians can share a beautiful mathematical vista with one another, but there is no camera that can easily capture an image of such a vista to convey it in full to people who have not trudged along many of the same trails.” (Silvio Levy, “The Eightfold Way: The Beauty of Klein’s Quartic Curve”, 1999)

About Mathematicians (1980-1989)

“When a mathematician asks himself why some result should hold, the answer he seeks is some intuitive understanding. In fact, a rigorous proof means nothing to him if the result doesn’t make sense intuitively.” (Morris Kline, “Mathematics: The Loss of Certainty”, 1980)

"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)

"[…] a mathematician's ultimate concern is that his or her inventions be logical, not realistic. This is not to say, however, that mathematical inventions do not correspond to real things. They do, in most, and possibly all, cases. The coincidence between mathematical ideas and natural reality is so extensive and well documented, in fact, that it requires an explanation. Keep in mind that the coincidence is not the outcome of mathematicians trying to be realistic - quite to the contrary, their ideas are often very abstract and do not initially appear to have any correspondence to the real world. Typically, however, mathematical ideas are eventually successfully applied to describe real phenomena […]"(Michael Guillen, "Bridges to Infinity: The Human Side of Mathematics", 1983)

"Mathematicians might actually be looking at life with a most trenchant sense - one that perceives things the other five senses cannot." (Michael Guillen, "Bridges to Infinity: The Human Side of Mathematics", 1983) 

“[…] mathematics is not a science – it is not capable of proving or disproving the existence of real things. In fact, a mathematician’s ultimate concern is that his or her inventions be logical, not realistic.” (Michael Guillen, “Bridges to Infinity: The Human Side of Mathematics”, 1983)

"Mathematics has been called the science of the infinite. Indeed, the mathematician invents finite constructions by which questions are decided that by their very nature refer to the infinite. This is his glory." (Hermann Weyl, "Axiomatic versus constructive procedures in mathematics" , The Mathematical Intelligencer, 1985)

"Mathematics is not a deductive science - that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork. You want to find out what the facts are, and what you do is in that respect similar to what a laboratory technician does. Possibly philosophers would look on us mathematicians the same way as we look on the technicians, if they dared." (Paul R Halmos, "I Want to be a Mathematician: An Automathography", 1985)

"There are many things you can do with problems besides solving them. First you must define them, pose them. But then of course you can also refi ne them, depose them, or expose them or even dissolve them! A given problem may send you looking for analogies, and some of these may lead you astray, suggesting new and different problems, related or not to the original. Ends and means can get reversed. You had a goal, but the means you found didn’t lead to it, so you found a new goal they did lead to. It’s called play. Creative mathematicians play a lot; around any problem really interesting they develop a whole cluster of analogies, of playthings." (David Hawkins, "The Spirit of Play", Los Alamos Science, 1987

About Mathematicians (1970-1979)

"Everything considered, mathematicians should have the courage of their most profound convictions and thus affirm that mathematical forms indeed have an existence that is independent of the mind considering them. […] Yet, at any given moment, mathematicians have only an incomplete and fragmentary view of this world of ideas." (René Thom, "Modern Mathematics: An Educational and Philosophical Error?", American Scientist Vol. 59, 1971)

"If some great mathematicians have known how to give lyrical expression to their enthusiasm for the beauty of their science, nobody has suggested examining it as if it were the object of an art - mathematical art -  and consequently the subject of a theory of aesthetics, the aesthetics of mathematics." (François Le Lionnais, "Great Currents of Mathematical Thought", 1971)

"Mathematicians are there to find the constraints and to eliminate those things that aren't constraints [...]" (Robert E Machol, Mathematicians are useful, 1971)

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

"For the mathematician, the physical way of thinking is merely the  starting point in a process of abstraction or idealization. Instead of being a dot on a piece of paper or a particle of dust suspended in space, a point becomes, in the mathematician's ideal way of thinking, a set of numbers or  coordinates. In applied mathematics we must go much further with this process because the physical problems under consideration are more complex. We first view a phenomenon in the physical way, of course, but we must then go through a process of idealization to arrive at a more abstract  representation of the phenomenon which will be amenable to mathematical analysis." (Peter Lancaster, "Mathematics: Models of the Real World", 1976)

"In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)

"You cannot read mathematics superficially; the inescapable abstraction always has an element of self-torture in it, and the one to whom this self-torture is joy is the mathematician." (Rózsa Péter, "Playing with Infinity: Mathematical Explorations and Excursions", 1976)

"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)

"Mathematicians have always been rather of a jealous nature, and undoubtedly jealousy was a family characteristic of the Bernoullis. There is some excuse for mathematicians, for their reputation stands for posterity largely not on what they did, but on what their contemporaries attributed to them." (Karl Pearson, "The History of Statistics in the 17th and 18th Century", 1978)

"On the face of it there should be no disagreement about mathematical proof. Everybody looks enviously at the alleged unanimity of mathematicians; but in fact there is a considerable amount of controversy in mathematics. Pure mathematicians disown the proofs of applied mathematicians, while logicians in turn disavow those of pure mathematicians. Logicists disdain the proofs of formalists and some intuitionists dismiss with contempt the proofs of logicists and formalists." (Imre Lakatos,"Mathematics, Science and Epistemology" Vol. 2, 1978)

"A proof is a construction that can be looked over, reviewed, verified by a rational agent. We often say that a proof must be perspicuous or capable of being checked by hand. It is an exhibition, a derivation of the conclusion, and it needs nothing outside itself to be convincing. The mathematician surveys the proof in its entirety and thereby comes to know the conclusion." (Thomas Tymoczko," The Four Color Problems", Journal of Philosophy , Vol. 76, 1979)

About Mathematicians (1960-1969)

"As every mathematician knows, nothing is more fruitful than these obscure analogies, these indistinct reflections of one theory into another, these furtive caresses, these inexplicable disagreements; also nothing gives the researcher greater pleasure." (André Weil, "De la Métaphysique aux Mathématiques", 1960)

"In fact, the construction of mathematical models for various fragments of the real world, which is the most essential business of the applied mathematician, is nothing but an exercise in axiomatics." (Marshall Stone, cca 1960)

"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)

"Nothing is more fruitful - all mathematicians know it - than those obscure analogies, those disturbing reflections of one theory on another; those furtive caresses, those inexplicable discords; nothing also gives more pleasure to the researcher. The day comes when this illusion dissolves: the presentiment turns into certainty; the yoked theories reveal their common source before disappearing. As the Gita teaches, one achieves knowledge and indifference at the same time. Metaphysics has become Mathematics, ready to form the material of some treatise whose cold beauty has lost the power to move us." (André Weil, "De la métaphysique aux mathématiques", 1960)

"It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better." (Paul Dirac, "The Evolution of the Physicist's Picture of Nature", 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)

"A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data. God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe." (Paul Dirac, Scientific American, 1963)

"[…] it is more important to have beauty in one's equations that to have them fit experiment. […] It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress." (Paul Dirac, Scientific American, 1963) 

"The mathematicians and physics men have their mythology; they work alongside the truth, never touching it; their equations are false But the things work. Or, when gross error appears, they invent new ones; they drop the theory of waves In universal ether and imagine curved space." (Robinson Jeffers," The Beginning and the End and Other Poems, The Great Wound", 1963) 

"It becomes the urgent duty of mathematicians, therefore, to meditate about the essence of mathematics, its motivations and goals and the ideas that must bind divergent interests together." (Richard Courant, "Mathematics in the Modern World", Scientific American Vol. 211, 1964)

"When the problems in physics become difficult we may often look to the mathematician who may already have studied such things and have prepared a line of reasoning for us to follow. On the other hand they may not have, in which case we have to invent our own line of reasoning, which we then pass back to the mathematician." (Richard Feynman,"The Character of Physical Law", 1965)

Marvin Minsky - Collected Quotes

"Computer languages of the future will be more concerned with goals and less with procedures specified by the programmer." (Marvin Minsky, "Form and Content in Computer Science", [Turing Award lecture] 1969)

"A memory should induce a state through which we see current reality as an instance of the remembered event - or equivalently, see the past as an instance of the present. […] the system can perform a computation analogous to one from the memorable past, but sensitive to present goals and circumstances." (Marvin Minsky, "K-Linesː A Theory of Memory", 1980) 

"Since we have no systematic way to avoid all the inconsistencies of commonsense logic, each person must find his own way by building a private collection of 'cognitive censors' to suppress the kinds of mistakes he has discovered in the past." (Marvin Minsky, "Jokes and their Relation to the Cognitive Unconscious", 1980) 

"When you, ‘get an idea’, or ‘solve a problem’, or have a ‘memorable experience’, you create what we shall call a K-line. This K-line gets connected to those ‘mental agencies’ that were actively involved in the memorable event. When that K-line is later ‘activated’, it reactivates some of those mental agencies, creating a ‘partial mental state’ resembling the original." (Marvin Minsky, "K-Linesː A Theory of Memory", 1980) 

"If explaining minds seems harder than explaining songs, we should remember that sometimes enlarging problems makes them simpler! The theory of the roots of equations seemed hard for centuries within its little world of real numbers, but it suddenly seemed simple once Gauss exposed the larger world of so-called complex numbers. Similarly, music should make more sense once seen through listeners' minds." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"The way the mathematics game is played, most variations lie outside the rules, while music can insist on perfect canon or tolerate a casual accompaniment." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"Theorems often tell us complex truths about the simple things, but only rarely tell us simple truths about the complex ones. To believe otherwise is wishful thinking or ‘mathematics envy’." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"What is the difference between merely knowing (or remembering, or memorizing) and understanding? […] A thing or idea seems meaningful only when we have several different ways to represent it - different perspectives and different associations […]. Then we can turn it around in our minds, so to speak: however it seems at the moment, we can see it another way and we never come to a full stop. In other words, we can 'think' about it. If there were only one way to represent this thing or idea, we would not call this representation thinking." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"The hardest problems we have to face do not come from philosophical questions about whether brains are machines or not. There is not the slightest reason to doubt that brains are anything other than machines with enormous numbers of parts that work in perfect accord with physical laws. As far as anyone can tell, our minds are merely complex processes. The serious problems come from our having had so little experience with machines of such complexity that we are not yet prepared to think effectively about them." (Marvin Minsky, 1986) 

"For generations, scientists and philosophers have tried to explain ordinary reasoning in terms of logical principles - with virtually no success. I suspect this enterprise failed because it was looking in the wrong direction: common sense works so well not because it is an approximation of logic; logic is only a small part of our great accumulation of different, useful ways to chain things together." (Marvin Minsky, "The Society of Mind", 1987) 

"Unless we can explain the mind in terms of things that have no thoughts or feelings of their own, we'll only have gone around in a circle." (Marvin Minsky, "The Society of Mind", 1987) 

"Every system that we build will surprise us with new kinds of flaws until those machines become clever enough to conceal their faults from us." (Marvin Minsky, "The Emotion Machine: Commonsense Thinking, Artificial Intelligence, and the Future of the Human Mind", 2006)

"It makes no sense to seek a single best way to represent knowledge - because each particular form of expression also brings its particular limitations. For example, logic-based systems are very precise, but they make it hard to do reasoning with analogies. Similarly, statistical systems are useful for making predictions, but do not serve well to represent the reasons why those predictions are sometimes correct." (Marvin Minsky, "The Emotion Machine: Commonsense Thinking, Artificial Intelligence, and the Future of the Human Mind", 2006)

"The idea that the world exists is like adding an extra term to an equation that doesn’t belong there." (Marvin Minsky)

About Mathematicians (1950-1959)

“Mathematicians create by acts of insight and intuition. Logic then sanctions the conquests of intuition. It is the hygiene that mathematics practice to keep its ideas healthy and strong. Moreover, the whole structure rests fundamentally on uncertain ground, the intuitions of man.” (Morris Kline, “Mathematics in Western Culture”, 1953)

"Mathematicians do not know what they are talking about because pure mathematics is not concerned with physical meaning. Mathematicians never know whether what they are saying is true because, as pure mathematicians, they make no effort to ascertain whether their theorems are true assertions about the physical world." (Morris Kline, “Mathematics in Western Culture”, 1953)

"The advantage is that mathematics is a field in which one’s blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is only one’s best moments that count and not one’s worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician’s reputation.” (Norbert Wiener, “Ex-Prodigy: My Childhood and Youth”, 1953)

"Mathematics, like music and poetry, is a creation of the mind; [...] the primary task of the mathematician, like that of any other artist, is to extend man's mental horizon by representation and interpretation." (Graham Sutton, "Mathematics in Action", 1954) 

"Rigor is to the mathematician what morality is to man. It does not consist in proving everything, but in maintaining a sharp distinction between what is assumed and what is proved, and in endeavoring to assume as little as possible at every stage." (André Weil, Mathematical Teaching in Universities", The American Mathematical Monthly Vol. 61 (1), 1954)

"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)

"You have to guess the mathematical theorem before you prove it: you have to guess the idea of the proof before you carry through the details. You have to combine observations and follow analogies: you have to try and try again. The result of the mathematician’s creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing" (George Polya, "Mathematics and plausible reasoning" Vol. 1, 1954)

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

"The mathematicians know a great deal about very little and the physicists very little about a great deal." (Stanislaw Ulam, "On the Ergodic Behavior of Dynamical Systems", 1955)

"A creative mathematician is the intersection of several unlikely events, For the most part we are ignorant of the nature of these events and of their probabilities. Some of them are at present quite beyond our controls. An example is the probability of a genetic composition necessary for intensive and highly abstract thinking. Others are clearly subject to our influence. For example, the probability of an early acquaintance with living mathematics and with the joy of mathematical achievement is determined by educational practices." (Kenneth O May "Undergraduate Research in Mathematics",  The American Mathematical Monthly 65, 1958) 

"Mathematical examination problems are usually considered unfair if insoluble or improperly described: whereas the mathematical problems of real life are almost invariably insoluble and badly stated, at least in the first balance. In real life, the mathematician's main task is to formulate problems by building an abstract mathematical model consisting of equations, which will be simple enough to solve without being so crude that they fail to mirror reality. Solving equations is a minor technical matter compared with this fascinating and sophisticated craft of model-building, which calls for both clear, keen common-sense and the highest qualities of artistic and creative imagination." (John Hammersley & Mina Rees, "Mathematics in the Market Place", The American Mathematical Monthly 65, 1958) 

“Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty's sake and pulls it down to earth.” (Marston Morse, 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)

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

About Mathematicians (1940-1949)

"A mathematician, like a painter or a poet, is a maker of patterns. [...]. The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics." (Godfrey H Hardy, “A Mathematician's Apology”, 1940)

"At the present time it is of course quite customary for physicists to trespass on chemical ground, for mathematicians to do excellent work in physics, and for physicists to develop new mathematical procedures […] Trespassing is one of the most successful techniques in science." (Wolfgang Köhler, "Dynamics in Psychology", 1940)

"It is a melancholic experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done [...] there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.“  (Godfrey H Hardy, "A Mathematician's Apology“, 1940)

“Mathematics is not a contemplative but a creative subject; no one can draw much consolation from it when he has lost the power or the desire to create; and that is apt to happen to a mathematician rather soon.” (Godfrey H Hardy, "A Mathematician's Apology“, 1940)

“The mathematician is still regarded as the hermit who knows little of the ways of life outside his cell, who spends his time compounding incredible and incomprehensible theorems in a strange, clipped, unintelligible jargon.” (James R Newman, “Mathematics and the Imagination”, 1940)

"[…] there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, [...] the mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real', but [...] [a physicist] is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics." (Godfrey H Hardy, "A Mathematician's Apology", 1940)

"We now come to a decisive step of mathematical abstraction: we forget about what the symbols stand for […] The mathematician] need not be idle; there are many operations which he may carry out with these symbols, without ever having to look at the things they stand for." (Hermann Weyl, "The Mathematical Way of Thinking", 1940)

"A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician. If this description were accurate, mathematics could not attract any intelligent person. It would be a game with definitions, rules and syllogisms, without motivation or goal." (Richard Courant & Herbert Robbins, "What Is Mathematics?", 1941)

"Mathematicians deal with possible worlds, with an infinite number of logically consistent systems. Observers explore the one particular world we inhabit. Between the two stands the theorist. He studies possible worlds but only those which are compatible with the information furnished by observers. In other words, theory attempts to segregate the minimum number of possible worlds which must include the actual world we inhabit. Then the observer, with new factual information, attempts to reduce the list further. And so it goes, observation and theory advancing together toward the common goal of science, knowledge of the structure and observation of the universe." (Edwin P Hubble, "The Problem of the Expanding Universe", 1941)

"Mathematicians themselves set up standards of generality and elegance in their exposition which are a bar to understand." (Kenneth E Boulding, "Economic Analysis", 1941)

"It is to be hoped that in the future more and more theoretical physicists will command a deep knowledge of mathematical principles; and also that mathematicians will no longer limit themselves so exclusively to the aesthetic development of mathematical abstractions." (George D Birkhoff, "Mathematical Nature of Physical Theories" American Scientific Vol. 31 (4), 1943)

"A mathematician is not a man who can readily manipulate figures; often he cannot. He is not even a man who can readily perform the transformations of equations by the use of calculus. He is primarily an individual who is skilled in the use of symbolic logic on a high plane, and especially he is a man of intuitive judgment in the choice of the manipulative processes he employs." (Vannevar Bush, "As We May Think", 1945)

"In various ways, methods of approaching the mathematician's ideal were sought, and the resulting suggestions were the source of much that was mistaken in metaphysics and theory of knowledge.” (Bertrand Russell, “A History of Western Philosophy”, 1945)

"One of the difficulties which a mathematician has in describing his work to non-mathematicians is that the present day language of mathematics has become so esoteric that a well educated layman, or even a group of scientists, can comprehend essentially nothing of the discourse which mathematicians hold with each other, or of the accounts of their latest researches which are published in their professional journals." (Angus E Taylor, "Some Aspects of Mathematical Research", American Scientist , Vol. 35, No. 2, 1947)

"[…] mathematicians progress only by doubt, through humble and constant attempts to impinge on the immense domain of the unknown." (Leopold Infeld, "Whom the Gods Love: The Story of Évariste Galois", 1948)

About Mathematicians (1910-1919)

“Mathematicians attach great importance to the elegance of their methods and their results. This is not pure dilettantism. What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details.” (Henri Poincaré, “The Future of Mathematics”, Monist Vol. 20, 1910)

"The critical mathematician has abandoned the search for truth. He no longer flatters himself that his propositions are or can be known to him or to any other human being to be true; and he contents himself with aiming at the correct, or the consistent. The distinction is not annulled nor even blurred by the reflection that consistency contains immanently a kind of truth. He is not absolutely certain, but he believes profoundly that it is possible to find various sets of a few propositions each such that the propositions of each set are compatible, that the propositions of each such set imply other propositions, and that the latter can be deduced from the former with certainty. That is to say, he believes that there are systems of coherent or consistent propositions, and he regards it his business to discover such systems. Any such system is a branch of mathematics." (Cassius J Keyser, Science, New Series, Vol. 35 (904), 1912)

"It may be surprising to see emotional sensibility invoked apropos of mathematical demonstrations which, it would seem, can interest only the intellect. This would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true esthetic feeling that all real mathematicians know, and surely it belongs to emotional sensibility." (Henri Poincaré, 1913)

"It would only be possible to imagine life or beauty as being strictly mathematical if we ourselves were such infinitely capable mathematicians as to be able to formulate their characteristics in mathematics so extremely complex that we have never yet invented them." (Theodore A Cook, "The Curves of Life", 1914)

“[…] the mathematician is always walking upon the brink of a precipice, for, no matter how many theorems he deduces, he cannot tell that some contradiction will not await him in the infinity of consequences.” (Richard A Arms, “The Notion of Number and the Notion of Class”, 1917)

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

About Mathematicians (1900-1909)

"The mathematician, carried along on his flood of symbols, dealing apparently with purely formal truths, may still reach results of endless importance for our description of the physical universe." (Karl Pearson, “The Grammar of Science”, 1900)

"Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change. Matter does not engage their attention, they are interested in form alone." (Henri Poincaré, "Science and Hypothesis", 1901)

"Mathematicians therefore proceed 'by construction', they 'construct' more complicated combinations. When they analyse these combinations, these aggregates, so to speak, into their primitive elements, they see the relations of the elements and deduce the relations of the aggregates themselves. The process is purely analytical, but it is not a passing from the general to the particular, for the aggregates obviously cannot be regarded as more particular than their elements." (Henri Poincaré, "Science and Hypothesis", 1902)

"The apodictic quality of mathematical thought, the certainty and correctness of its conclusions, are due, not to a special mode of ratiocination, but to the character of the concepts with which it deals. What is that distinctive characteristic? I answer: precision, sharpness, completeness of definition. But how comes your mathematician by such completeness? There is no mysterious trick involved; some ideas admit of such precision, others do not; and the mathematician is one who deals with those that do." (Cassius J Keyser, "The Universe and Beyond", Hibbert Journal Vol. 3, 1904–1905)

"To think the thinkable - that is the mathematician's aim." (Cassius J Keyser, "The Universe and Beyond", Hibbert Journal Vol. 3, 1904-1905)

“A mathematical theorem and its demonstration are prose. But if the mathematician is overwhelmed with the grandeur and wondrous harmony of geometrical forms, of the importance and universal application of mathematical maxims, or, of the mysterious simplicity of its manifold laws which are so self-evident and plain and at the same time so complicated and profound, he is touched by the poetry of his science; and if he but understands how to give expression to his feelings, the mathematician turns poet, drawing inspiration from the most abstract domain of scientific thought.” (Paul Carus, “Friedrich Schiller”, 1905)

"The chief end of mathematical instruction is to develop certain powers of the mind, and among these the intuition is not the least precious. By it the mathematical world comes in contact with the real world, and even if pure mathematics could do without it, it would always be necessary to turn to it to bridge the gulf between symbol and reality. The practician will always need it, and for one mathematician there are a hundred practicians. However, for the mathematician himself the power is necessary, for while we demonstrate by logic, we create by intuition; and we have more to do than to criticize others’ theorems, we must invent new ones, this art, intuition teaches us." (Henri Poincaré, "The Value of Science", 1905)

"The great mathematician, like the great poet or great naturalist or great administrator, is born." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)