If fuzzy logic is construed, as Zadeh and co. suggest it should be, as a nonclassical theory of truth-preserving inferences, fuzzy technology does not rely on it, and so the successes of that technology cannot be claimed to its credit. If, on the other hand, fuzzy logic is construed as an attempt to represent the mental processes through which people go when making adjustments to kiln thermostats, air-conditionaers, etc., there is a connection with fuzzy technology. But, of course, so construed, fuzzy logic is not, after all, an attempt to represent truth-preserving inferences, and is not, after all, a theory in the same domain as classical logic; in fact, so construed, it is obviously not properly describable as a "logic" at all. | |

Susan Haack, from Deviant Logic, Fuzzy Logic, 1996. |
756 |

While, traditionally, logic has corrected or avoided it, fuzzy logic | |

Susan Haack, from Deviant Logic, Fuzzy Logic, 1996. |
755 |

Fuzzy logic lacks every feature that the pioneers of modern logic wanted logic | |

Susan Haack, from Deviant Logic, Fuzzy Logic, 1996. |
754 |

The shortest path between two assertions about the reals passes through the complexes. | |

J. Hadamard, quoted in Theory of Complex Functions, by Reinhold Remmert. |
148 |

Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle. | |

Jacques Hadamard, quoted in Euclidean and Non-Euclidean Geometries by Greenberg |
875 |

Anyone, anywhere along the line, can fill in the details and check them. The fact that a computer can run through more details in a few hours than a human could ever hope to do in a lifetime does not change the basic concept of mathematical proof. What has changed is not the theory but the practice of mathematics. | |

Wofgang Haken, quoted in From Here to Infinity, by Ian Stewart. |
585 |

The universe is not only queerer than we suppose but queerer than we can suppose. | |

J.B.S. Haldane, quoted in Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise, by Manfred Schroder. |
566 |

The only way to learn mathematics is to do mathematics. That tenet is the foundation of the do-it-yourself, Socratic, or Texas method, the method in which the teacher plays a role of an omniscient but largely uncommunicative referee between the learner and the facts. | |

Paul Halmos, quoted in Out of the Mouths of Mathematicians, by Rosemary Schmalz |
450 |

It's hard for me to get used to the absence of pressure. I always pushed myself under pressure, and of course, I blamed the world. The world is putting on the pressure. Well, now I'm beginning to realize that the world is not putting on the pressure. If I never published anything, not even an elementary textbook, if I never again answered a letter, if I never did anything any more except drink my beer and watch the telly, nobody would, I think, think any the worse of me. But I keep putting myself under a little pressure and keep doing these small piddling jobs. | |

Paul Halmos, quoted in "In touch with God: An interview with Paul Halmos," by Don Albers, College Mathematics Journal, vol. 35, no. 1, January 2004, pp. 2- 14 |
868 |

Learning mathematics is always extraordinarily hard work. I can't easily read mathematics. I can't listen to lectures. The only thing I enjoy is a kind of mathematical gossip, when people sit in easy chairs with their feet up on something and tell me their mathematics; then I can learn. | |

Paul Halmos, quoted in "In touch with God: An interview with Paul Halmos," by Don Albers, College Mathematics Journal, vol. 35, no. 1, January 2004, pp. 2- 14 |
867 |

Mathematics is not a deductive science -- that's a cliche. 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. | |

Paul R. Halmos, from I Want to Be a Mathematician |
1148 |

It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts. | |

Paul Halmos, quoted in Out of the Mouths of Mathematicians, by Rosemary Schmalz |
451 |

A smooth lecture... may be pleasant; a good teacher challenges, asks, irritates and maintains high standards - all that is generally not pleasant. | |

Paul Halmos, |
151 |

Teachers of elementary mathematics in the USA frequently complain that all calculus books are bad. That is a case in point. Calculus books are bad because there is no such subject as calculus; it is not a subject because it is many subjects. What we call calculus nowadays is the union of a dab of logic and set theory, some axiomatic theory of complete ordered fields, analytic geometry and topology, the latter in both the "general" sense (limits and continuous functions) and the algebraic sense (orientation), real-variable theory properly so called (differentiation), the combinatoric symbol manipulation called formal integration, the first steps of low-dimensional measure theory, some differential geometry, the first steps of the classical analysis of the trigonometric, exponential, and logarithmic functions, and, depending on the space available and the personal inclination of the author, some cook-book differential equations, elementary mechanics, and a small assortment of applied mathematics. Any one of these is hard to write a good book on; the mixture is impossible. | |

Paul Halmos, quoted in Excursions in Calculus, by Robert Young. |
152 |

Mathematics - this may surprise or shock some - is never deductive in its creation. The mathematician at work makes vague guesses, visualizes broad generalizations, and jumps to unwarranted conclusions. He arranges and rearranges his ideas, and he becomes convinced of their truth long before he can write down a logical proof... The deductive stage, writing the result down, and writing its rigorous proof are relatively trivial once the real insight arrives; it is more like the draftsman's work not the architect's. | |

Paul Halmos, quoted in Out of the Mouths of Mathematicians, by R. Schmalz. |
150 |

A good stack of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one. | |

Paul Halmos, quoted in Contemporary Abstract Algebra, by J. Gallian. |
149 |

To be able to be caught up into the world of thought - that is educated. | |

Edith Hamilton, quoted in The Beacon Book of Quotations by Women, edited by Rosalie Maggio. |
153 |

This above all: To thine own self be true, And it must follow, as the night the day, Thou canst not then be false to any man. | |

Hamlet, |
936 |

When the morning's freshness has been replaced by the weariness of midday, when the leg muscles quiver under the strain, the climb seems endless, and suddenly nothing will go quite as you wish - it is then that you must not hesitate. | |

Dag Hammarskjold, |
1310 |

The purpose of computing is insight, not numbers! | |

R. W. Hamming, quoted in "Mathematics and Modern Technology" by R.S. Pinkham, in The American Mathematical
Monthly, vol. 103, no. 7, Aug.-Sept.1996, pp. 539-545. |
154 |

People usually consider walking on water or in thin air a miracle. But I think the real miracle is not to walk either on water or in thin air, but to walk on earth. Every day we are engaged in a miracle which we don't even recognize: a blue sky, white clouds, green leaves, the black, curious eyes of a child -- our own two eyes. All is a miracle. | |

Thich Nhat Hanh, |
1018 |

Paul Erdos has a theory that God has a book containing all the theorems of mathematics with their absolutely most beautiful proofs, and when he wants to express particular appreciation of a proof he exclaims, 'This is from the book!' | |

Ross Hansberger, quoted in Out of the Mouths of Mathematicians, by R. Schmalz. |
155 |

Hardy's New Year's Resolutions:
1. Prove the Riemann hypothesis. 2. Make 211 no out in the fourth innings of the last test match at the Oval. 3. Find an argument for the nonexistence of God which shall convince the general public. 4. Be the first man at the top of Mt. Everest. 5. Be proclaimed the first president of the U.S.S.R., of Great Britian, and Germany. 6. Murder Mussolini. 3. Find an argument for the nonexistence of God which shall convince the general public. 4. Be the first man at the top of Mt. Everest. 5. Be proclaimed the first president of the U.S.S.R., of Great Britian, and Germany. 6. Murder Mussolini. | |

G.H. Hardy, from The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel |
882 |

Here was a man who could work out modular equations and theorems of complex multiplication, to orders unheard of, whose mastery of continued fractions was, on the formal side at any rate, beyond that of any mathematician in the world, who had found for himself the functional equation of the Zeta-function, and the dominant terms of many of the most famous problems in the analytic theory of numbers; and he had never heard of a doubly periodic function or of Cauchy's theorem, and he had indeed but the vaguest idea of what a function of a complex variable was. His ideas as to what constituted a mathematical proof were of the most shadowy description. All his results, new or old, right or wrong, had been arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account. | |

G.H. Hardy, from The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel |
880 |

I have [very rarely] encountered a pupil who could face the simplest problem involving the ideas of infinity, limit, or continuity with a vestige of the confidence with which he could deal with questions of a different character and of far greater intrinsic difficulty. | |

G.H. Hardy, from The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel |
879 |

He [Ramanujan] would probably have been a greater mathematician if he had been caught and tamed a little in his youth; he would have discovered more that was new, and that, no doubt, of greater importance. On the other hand he would have been less of a Ramanujan, and more of a European professor, and the loss might have been greater than the gain. He [Ramanujan] would probably have been a greater mathematician if he had been caught and tamed a little in his youth; he would have discovered more that was new, and that, no doubt, of greater importance. On the other hand he would have been less of a Ramanujan, and more of a European professor, and the loss might have been greater than the gain. | |

G.H. Hardy, from The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel |
883 |

The positive integers stand there, a continual and inevitable challenge to the curiosity of every healthy mind. | |

G.H. Hardy, quoted in Elementary Number Theory by David M. Burton. |
905 |

I think that it is time that teachers of geometry become a little more ambitious… It seems to me regrettable that students are not given the opportunity, while still at school, of learning a good deal more about the real subject matter out of which modern geometrical systems are built. It is probably easier, and certainly vastly more instructive than a great deal of what they are actually taught… I have not yet encountered a student who finds difficulty with such ideas [projective geometry, the nature of axiom systems, and perspective]… [We must] widen the horizon of knowledge, recognizing, as regards the niceties of logic, sequence, and exposition, that the elementary geometry of schools is a fundamentally and inevitably illogical subject. | |

G.H. Hardy, from "What is Geometry?" Hardy's Presidential Address to The Mathematical Association in 1925. Reprinted in The Changing Shape of Geometry edited by Chris Pritchard, Cambridge University Press, 2003. |
1788 |

The mathematician is in much more direct contact with reality [than the physicist]. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real'; but a very little reflection is enough to show that the physicist's reality, whatever it may be, has few or none of the attributes which common sense ascribes instinctively to reality. A chair may be a collection of whirling electrons, or an idea in the mind of God: each of these accounts of it may have its merits, but neither conforms at all closely to the suggestions of common sense.
… Neither the physicists nor philosophers have ever given any convincing account of what 'physical reality' is, or of how the physicist passes, from the confused mass of fact or sensation with which he starts, to the construction of the objects which he calls 'real'. Thus we cannot be said to know what the subject-matter of physics is; but this need not prevent us from understanding roughly what a physicist is trying to do. It is plain that he is trying to correlate the incoherence body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme which he can borrow only from mathematics. A mathematician, on the other hand, is working with his own mathematical reality… A chair or a star is not in the least like what it seems to be; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it; but '2' or '317' has nothing to do with sensation, and its properties stand out the more clearly the more closely we scrutinize it. It may be that modern physics fits best into some framework of idealistic philosophy… Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is prime, not because we think so, or because our minds are shaped in one way rather than another, but | |

G.H. Hardy, from "A Mathematician's Apology," 1992 Canto edition.(p.128-130) |
1538 |

We have done no more than to make explicit a few of the instinctive prejudices of the 'mathematician in the street.' Yet with our first demand [mathematical truth is immutable and unconditional] we have antagonized at least two-thirds of the philosophers in the world; and with the second [mathematical theorems are statements about reality] we have reduced our first indiscretion to entire insignificance, since we have committed ourselves, in one form or another, to the objective reality of propositions, a doctrine rejected, I believe, not only by all philosophers, but also by all three of the current schools of mathematical logic. | |

G.H. Hardy, quoted in Musings of the Masters edited by Raymond G. Ayoub. |
1472 |

Stellar astronomy and atomic physics are the only sciences which stand higher in popular estimation [than mathematics]. A mathematician need not consider himself on the defensive… The public does not need to be convinced that there is something in mathematics. | |

G.H. Hardy, from "A Mathematician's Apology," 1992 Canto edition.(p.64-65) |
1529 |

Chess problems are the hymn-tunes of mathematics. We may learn the same lesson, at a lower level but for a wider public, from bridge, or descending further, from the puzzle columns of the popular newspapers. Nearly all of their immense popularity is a tribute to the drawing power of rudimentary mathematics… What the public wants is a little intellectual 'kick', and nothing else has quite the kick of mathematics. | |

G.H. Hardy, from "A Mathematician's Apology," 1992 Canto edition.(p.87-88) |
1537 |

Yet a chess problem is | |

G.H. Hardy, from "A Mathematician's Apology," 1992 Canto edition.(p.87) |
1536 |

It would be difficult now to find an educated man quite insensitive to the aesthetic appeal of mathematics… The fact is that there are few more 'popular' subjects than mathematics. Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances may suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable, whereas most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity. | |

G.H. Hardy, from "A Mathematician's Apology," 1992 Canto edition.(p.86-87) |
1535 |

Practically all substantial contributions to human happiness have been made by ambitious men. | |

G.H. Hardy, from "A Mathematician's Apology," 1992 Canto edition.(p.78) |
1534 |

What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men. | |

G.H. Hardy, from "A Mathematician's Apology," 1992 Canto edition.(p.76) |
1533 |

It is quite true that most people can do nothing well. If so, it matters very little what career they choose. | |

G.H. Hardy, from "A Mathematician's Apology," 1992 Canto edition.(p.73) |
1532 |

I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, the least different to the amenity of the world. I have helped to train other mathematicians, but mathematicians of the same kind as myself, and their work has been, so far at any rate as I have helped them to it, as useless as my own. Judged by all practical standards, the value of my mathematical life is nil; and outside of mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. | |

G.H. Hardy, from "A Mathematician's Apology," 1992 Canto edition.(p.150-151) |
1540 |

Good work is not done by 'humble' men. | |

G.H. Hardy, from "A Mathematician's Apology," 1992 Canto edition.(p.66) |
1530 |

There is one comforting conclusion which is easy for a real mathematician. Real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years. | |

G.H. Hardy, from "A Mathematician's Apology," 1992 Canto edition.(p.140) |
1539 |

If then I find myself writing, not mathematics, but 'about' mathematics, it is a confession of weakness, for which I may rightly be scorned or pitied by younger and more vigorous mathematicians. I write about mathematics because, like any other mathematician who has passed sixty, I have no longer the freshness of mind, the energy, or the patience to carry on effectively with my proper job. | |

G.H. Hardy, from "A Mathematician's Apology," 1992 Canto edition.(p.63) |
1528 |

It is a melancholy 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… Exposition, criticism, appreciation, is work for second-rate minds. | |

G.H. Hardy, from "A Mathematician's Apology," 1992 Canto edition.(p. 61) |
1527 |

I believe that mathematical reality lies outside of us and that our function is to discover, or observe it, and that the theorems which we prove, and which we describe gradiloquently as our "creations" are simply notes on our observations. | |

G. H. Hardy, |
156 |

The elementary theory of numbers should be one of the very best subjects for early mathematical instruction. It demands very little previous knowledge; its subject matter is tangible and familiar; the processes of reasoning which it employs are simple, general and few; and it is unique among the mathematical sciences in its appeal to natural human curiosity. A month's intelligent instruction in the theory of numbers ought to be twice as instructive, twice as useful, and at least ten times as entertaining as the same amount of "calculus for engineers.'' | |

G. H. Hardy, quoted in Excursions in Calculus, by Robert Young. |
157 |

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. | |

G. H. Hardy, |
158 |

…Most people can do nothing at all well. | |

G.H. Hardy, from "A Mathematician's Apology," 1992 Canto edition.(p.67) |
1531 |

A person needs at intervals to separate from family and companions and go to new places. One must go without familiars in order to be open to influences, to change. | |

Katharine Butler Hathaway, |
1364 |

A continuous curve can fill a portion of space: this is one of the most remarkable facts of set theory, whose discovery we owe to G. Peano. | |

Hausdorff, quoted in Analysis by Its History by E. Hairer and G. Wanner. |
1125 |

If you do not make mistakes then you are not trying. | |

Coleman Hawkins, quoted in " Practice almost makes perfect" by Jean Mastrangeli, PRIMUS, vol. XI, no. 4, Dec. 2001, pp. 337-47. |
915 |

Easy reading is damned hard writing. | |

Nathaniel Hawthorne, quoted in A Primer of Mathematical Writing by Steven G. Krantz. |
740 |

Life is never so bad at its worst that it is impossible to live; it is never so good at it's best that it is easy to live. | |

Gabriel Heatter, |
932 |

The major lesson of all the wars is the fact that nobody takes any lesson out of them. | |

Hegel, |
1394 |

Music is architecture translated or transposed from space into time; for in music, besides the deepest feeling, there reigns also a rigorous mathematical intelligence. | |

Georg Hegel, from A Dictionary of Quotations in Mathematics by Nowlan |
1668 |

The Nothing Nots. | |

M. Heidegger, quoted in Sweet Reason, by J. Henle and T. Tymoczko. |
159 |

Problems worthy of attack prove their worth by fighting back. | |

Piet Hein, |
1409 |

…The definition of irrational numbers, on which geometric representations have often had a confusing influence… I take in my definition a purely formal point of view, calling some given symbols numbers, so that the existence of these numbers is beyond doubt. | |

Heine, quoted in Analysis by Its History by E. Hairer and G. Wanner. |
1112 |

Where books are burned, in the end people will be burned. | |

Heinrich Heine, from the U.S. Holocaust Memorial Museum |
969 |

Whenever they burn books they will also, in the end, burn human beings. | |

Heinrich Heine, |
998 |

I am free because I know that I alone am morally responsible for everything I do. | |

Robert Heinlein, |
844 |

We have to remember that what we observe is not nature in itself but nature exposed to our method of questioning. | |

Werner Heisenberg, from Physics and Philosophy. |
1059 |

In my paper the fact the XY was not equal to YX was very disagreeable to me. I felt this was the only point of difficulty in the whole scheme...and I was not able to solve it. | |

W. Heisenberg, from Contemporary Abstract Algebra, by J. Gallian. |
160 |

In this way quantum theory reminds us, as Bohr has put it, of the old wisdom that when searching for harmony in life one must never forget that in the drama of existence we are ourselves both players and spectators. It is understandable that in our scientific relation to nature our own activity becomes very important when we have to deal with parts of nature into which we can penetrate only by using the most elaborate tools. | |

Werner Heisenberg, from Physics and Philosophy. |
1060 |

In the natural sciences, then, the object of research is no longer nature as such, but a nature confronted by human questions, and in this sense, here too, man encounters himself. | |

Werner Heisenberg, Quoted in Radical Constructivism: A Way of Knowing and Learning by Ernst von Glasersfeld. |
773 |

We have to remember that what we observe is not nature itself, but nature exposed to our methods of questioning. | |

Wener Heisenberg, quoted in Teaching as a Subversive Activity, p. 78. |
1399 |

At first, I was deeply alarmed. I had the feeling that, through the surface of atomic phenomena, I was looking at the strangely beautiful interior and felt almost giddy at the thought that I now had to probe this wealth of mathematical structures spread out before me. | |

Werner Heisenberg, quoted in Insights of Genius: Imagery and Creativity in Science and Art, by Arthur I. Miller. |
563 |

What we observe is not nature itself, but nature exposed to our method of questioning. | |

Werner Heisenberg, |
1283 |

Orr would be crazy to fly more missions and sane if he didn't,but if he was sane he had to fly them. If he flew them he was crazy and didn't have to; but if he didn't want to he was sane and had to. | |

Joseph Heller, from Words I Wish I Wrote by Robert Fulgham |
950 |

All good books are alike in that they are truer than if they had really happened and after you are finished reading one you will feel that all that happened to you and afterwards it all belongs to you; the good and the bad, the ecstasy, the remorse, and sorrow, the people and the places and how the weather was. | |

Ernest Hemingway, |
999 |

With the aid of hyperspace philosophy, Theosophy, fantasies like Abbott's Flatland, and the science fiction of Wells and others, the fourth dimension had become almost a household word by 1910. Non-Euclidean geometry never achieved such a widespread popularity, in part because it did not lend itself to such a variety of interpretations. Ranging from an ideal Platonic or Kantian reality -- or even Heaven -- to the answer to all of the problems puzzling contemporary science, the fourth dimension could be all things to all people. | |

Linda Dalrymple Henderson, from The Fourth Dimension and non-Euclidean Geometry in Modern Art. |
663 |

When understood in their original context, however, "the fourth dimension" and no-Euclidean geometry are far from being the "scourge of every history of modern painting," as they have been termed. Instead, these concepts open the door to our understanding more fully the goals of many seminal artists of the early twentieth century. | |

Linda Dalrymple Henderson, from The Fourth Dimension and non-Euclidean Geometry in Modern Art. |
662 |

In most sciences one generation tears down what another has built and what one has established another undoes. In mathematics alone each generation adds a new story to the old structure. | |

Herman Henkel, from A Mathematical Journey, by S. Gudder. |
161 |

The struggle to become a better teacher begins all over again with the advent of each new class. | |

Martin Henley, |
162 |

The function of education has never been to free the mind and the spirit of man, but to bind them; and to the end that the mind and spirit of his children should never escape, Homo Sapiens has employed praise, ridicule, admonition, accusation, mutilation, and even torture to chain them to the culture pattern. | |

Jules Henry, quoted in Teaching Is..., by Merrill Harmin and Tom Gregory. |
163 |

The paradox of the human condition is expressed more in education than elsewhere in human culture, because learning to learn has been and continues to be Homo Sapiens' most formidable evolutionary task... It must also be clear that we will never quite learn how to learn, for since Homo Sapiens is self-changing, and since the more culture changes the faster it changes, man's methods and rate of learning will never quite keep pace with his need to learn. | |

Jules Henry, quoted in Teaching Is..., by Merrill Harmin and Tom Gregory. |
164 |

If you do not expect the unexpected, you will not find it; for it is hard to be sought out, and difficult. | |

Heraclitus, quoted in Mathematics and the Imagination by Edward Kasner and James Newman. |
689 |

I find mathematics an infinitely complex and mysterious world; exploring it is an addiction from which I hope never to be cured. | |

Reuben Hersh, from The Mathematical Experience |
861 |

The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver. | |

I. N. Herstein, quoted in Out of the Mouths of Mathematicians, by R. Schmalz. |
166 |

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

I. N. Herstein, from Contemporary Abstract Algebra, by J. Gallian. |
165 |

One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than we originally put into them. | |

Heinrich Hertz, |
167 |

... I do not consider myself less ignorant than most people. I have been and still am a seeker, but I have ceased to question stars and books; I have begun to listen to the teachings my blood whispers to me. My story is not a pleasant one; it is neither sweet nor harmonious, as invented stories are; it has the taste of nonsense and chaos, of madness and dreams - like the lives of all men who stop deceiving themselves. | |

Hermann Hesse, from the Prologue of Demain |
1062 |

What Weyl and Brouwer are doing is none other than a revival of Kronecker"s idea. They try to save mathematics by tossing overboard all that provokes concern... They crumble and chop science. If we accepted the reform they propose, then we would run the risk of losing the greatest part of our precious treasure. | |

David Hilbert, quoted in In Search of Infinity by N.Ya. Vilenkin (translated by Abe Shenitzer). |
728 |

Before beginning [to try to prove Fermat's Last Theorem] I should have to put in three years of intensive study, and I haven't that much time to squander on a probable failure. | |

David Hilbert, quoted in "Fermat's Last Stand," by Simon Singh and Kenneth A. Ribet, in Scientific American, November 1997. |
599 |

We have already seen that the infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to. Can though about things be so much different from things? Can thinking processes be so unlike the actual process of things? In short, can thought be so far removed from reality? Rather is it not clear that, when we think that we have encountered the infinite in some real sense, we have merely been seduced into thinking so by the fact that we often encounter extremely large and extremely small dimensions in reality? | |

David Hilbert, quoted in Understanding the Infinite by Shaughan Lavine. |
547 |

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

David Hilbert, |
1495 |

This conviction... is a powerful incentive. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no "we shall not know." | |

David Hilbert, quoted in Bridges to Infinity by Michael Guillen. |
535 |

One must admit that the state we are in now vis-a-vis the paradoxes is in the long run unendurable. Just think of it: in mathematics, this standard of trustworthiness and truth - the forming of concepts of inferences, as learned, taught, and used by all of us, can lead to nonsense. Where is one to find reliability and truth if mathematical thought can fail? | |

David Hilbert, quoted in In Search of Infinity by N. Ya Vilenkin |
1492 |

[On Cantor's work:] The finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity. | |

David Hilbert, quoted in The History of Mathematics, by D. Burton. |
170 |

Mathematics is an organism for whose vital strength the indissoluble union of the parts is a necessary condition. | |

David Hilber, quoted in Mathematical Thought from Ancient to Modern Times by Morris Kline. |
1093 |

With his theorem, which states that a continuous function of a real variable actually attains its least upper and greatest lower bounds, i.e., necessarily possesses a maximum and a minimum, Weierstrass created a tool which today is indispensable to all mathematicians for more refined analytical or arithmetical investigations. | |

Hilbert, quoted in Analysis by Its History by E. Hairer and G. Wanner. |
1114 |

In mathematics, as in any scientific research, we find two tendencies present. On the one hand, the tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material that is being studied, and to correlate the material in a systematic and orderly manner. On the other hand the tendency toward intuitive understanding fosters a more immediate group of the subjects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations. | |

David Hilbert, quoted in Out of the Mouths of Mathematicians, by R. Schmalz. |
168 |

Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. | |

David Hilbert, quoted in Excursions in Calculus, by Robert Young. |
171 |

No one will expel us from the paradise that Cantor has created. | |

David Hilbert, quoted in Patterns in Mathematics, by McCowen and Sequeira. |
169 |

Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. | |

David Hilbert, quoted in Excursions in Calculus, by R.M. Young. |
173 |

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

David Hilbert, from Geometry and the Imagination |
815 |

The infinite! No other question has ever moved so profoundly the spirit of man. | |

David Hilbert, |
172 |

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

David Hilbert, from Geometry and the Imagination. |
1242 |

The conception of the inconceivable [imaginary], this measurement of what not only does not, but cannot exist, is one of the finest achievements of the human intellect. No one can deny that such imaginings are indeed imaginary. But they lead to results grander than any which flow from the imagination of the poet. The imaginary calculus is one of the masterkeys to physical science. These realms of the inconceivable afford in many places our only mode of passage to the domains of positive knowledge. Light itself lay in darkness until this imaginary calculus threw light upon light. And in all modern researches into electricity, magnetism, and heat, and other subtile physical inquiries, these are the most powerful instruments. | |

Thomas Hill, quoted in Memorabilia Mathematica, by Robert E. Moritz. |
175 |

A pamphlet is never read more than once, but a song is learned by heart and repeated over and over again. | |

Joe Hill, |
1233 |

The discoveries of Newton have done more for England and for the race, than has been done by whole dynasties of British monarchs. | |

Thomas Hill, quoted in The History of Mathematics, by D. Burton. |
174 |

The U.S. Bureau of Labor Statistics predicted that OR [Operations Research] will be the third fastest- growing career area for U.S. college graduates from 1990 to 2005. It is also predicted that 100,000 people will be employed as operations research analysts in the United States by the year 2005. | |

F. Hillier and G. Lieberman, from Introduction to Operations Research. |
176 |

Mathematics should be fun. | |

Peter J. Hilton, from "Avoiding Math Avoidance," in Mathematics Tomorrow, by Lynn Arthur Steen. |
556 |

Particularly perverse and absurd is the multiple-choice format. I have been doing mathematics now as a professional for nearly 40 years and have never met a situation (outside of finite group theory!) in which I was faced with a mathematical problem and knew that the answer was one of five possibliites. Moreover if faced, artifically, by such a situation, may approach would, and should, be quite different from that in which I simply had to solve the problem. | |

Peter J. Hilton, from "Avoiding Math Avoidance," in Mathematics Tomorrow, by Lynn Arthur Steen. |
554 |

Tests tyrannize us -- they tyrannize teachers and children. They loom so large that they distort the teaching curriculum and the teacher"s natural style; they occur so frequently, and with such dire consequences, that they appear to the child (and, perhaps, to the teacher) to be the very reason for learning mathematics.
| |

Peter J. Hilton, from "Avoiding Math Avoidance," in Mathematics Tomorrow, by Lynn Arthur Steen. |
555 |

I.J. Good, a wartime colleague and friend, has aptly remarked that it is forunate that the authorities did not know during the war that [Alan] Turing was a homosexual; otherwise, the Allies might have lost the war. | |

Peter Hilton, from "Cryptanalysis in World War II -- and Mathematics Education," Mathematics Teacher, Oct. 1984. |
574 |

No wonder that Churchill described this effort [the British codebreakers working at Bletchley Park] as "Britian"s secret weapon," a weapon far more effective than the buzz bombs and the rockets that Werner von Braun designed for a German victory, a weapon absolutely decisive, in the judgement of many, in winning the war for the Allies. | |

Peter Hilton, from "Cryptanalysis in World War II -- and Mathematics Education," Mathematics Teacher, Oct. 1984. |
575 |

Just as any sensitive human being can be brought to appreciate beauty in art, music or literature, so that person can be educated to recognize the beauty in a piece of mathematics. The rarity of that recognition is not due to the "fact" that most people are not mathematically gifted but to the crassly utilitarian manner of teaching mathematics and of deciding syllabi and curricula, in which tedious, routine calculations, learned as a skill, are emphasized at the expense of genuinely mathematical ideas, and in which students spend almost all their time answering someone else's questions rather than asking their own. | |

Peter Hilton, from "Review: The Pleasures of Counting", American Mathematical Monthly, vol. 105, no. 5, May 1998. |
700 |

In March of 1945, I became a soldier and in April a prisoner of war, surviving in the meadows along the Rhine, always under the open sky in rain and sunshine, scribbling mathematics on toilet paper, the only paper available… To receive ration cards, I had to clean the British barracks. But on the first day, a British soldier asked me in fluent German what I was doing there and what I actually wanted to do. I said, "Mathematics!" He put me in his jeep and drove me home: "Study mathematics!" I did. To this day, I regret that I did not ask his name. | |

Friedrich E. Hirzebruch, |
1513 |

When we say that anything is infinite, we signify only that we are not able to conceive the ends and bounds of the thing named. | |

Thomas Hobbes, quoted in To Infinity and Beyond by Eli Maor. |
645 |

To understand this [Torricelli's Trumpet, a.k.a. Gabriel's Horn] for sense, it is not required that a man should be a geometrician or a logician, but that he should be mad. | |

Thomas Hobbes, quoted in "Torricelli"s Infinitely Long Solid and Its Philosophical Reception in the Seventeenth Century," by P. Mancosu and E. Vailati, Isis, vol. 82, 1991. |
620 |

Numbers as realities misbehave. | |

Douglas Hofstadter, from Godel, Escher, Bach: An Eternal Golden Braid. |
719 |

He told me there was no past. Or should I say, that the past is always on and waiting. Memories coded on a billion chemical loops playing the background like a movie with the sound turned down, until we reach back for that day in time. That face. That kiss. He said, Oh, our marvelous, mysterious little brains. Evolved from a single-celled bacteria we once were, huddling together for survival, and we still communicate as those colonies once did, one impulse colliding with the next impulse, in lightning strikes of fight or flight, leaping the gaps of synapse; and the memories we retrieve most frequently carve the deepest neural pathways, like water eroding a mountain gorge, and become our truth. | |

Sheri Holman, spoken by the character Eddie Alley in the book Witches on the Road Tonight. |
1807 |

My mind rebels at stagnation. Give me problems, give me work, give me the most abstruse cryptogram, or the most intricate analysis, and I am in my own proper atmosphere. But I abhor the dull routine of existence. I crave for mental exaltation. | |

Sherlock Holmes, from the character by Sir Arthur Conan Doyle. |
1251 |

The best of a book is not the thought which it contains, but the thought which it suggests; just as the charm of music dwells not in the tones but in the echoes of our hearts. | |

Oliver Wendell Holmes, |
1000 |

The freeman, casting with unpurchased hand, the vote that shakes the turrets of the land. | |

Oliver Wendell Holmes, |
1367 |

One's mind, once stretched by a new idea, never regains its original dimensions. | |

Oliver Wendell Holmes, |
1368 |

A new and valid idea is worth more than a regiment and fewer men can furnish the former than can command the later. | |

Oliver Wendel Holmes, Jr, quoted in A Teacher's Treasury of Quotations, |
177 |

Your education begins when what is called your education ends. | |

Oliver Wendell Holmes, Jr., |
1160 |

What a school thinks about its library is a measure of what it feels about education. | |

Harold Howe, |
1001 |

Fences are for those who cannot fly. | |

Elbert Hubbard, |
1589 |

When on the brink of complete discouragement, success is discerning that... the line between failure and success is so fine that often a single extra effort is all that is needed to bring victory out of defeat. | |

Elbert Green Hubbard, |
1587 |

We work to become, not to acquire. | |

Elbert Hubbard, from 101 Epigrams According to Elbert Hubbard, Roycroft Press |
1526 |

Success: A constant sense of discontent, broken by brief periods of satisfaction on doing some especially good piece of work. | |

Elbert Hubbard, from 101 Epigrams According to Elbert Hubbard, Roycroft Press |
1525 |

Happiness is a habit - cultivate it. | |

Elbert Hubbard, from 101 Epigrams According to Elbert Hubbard, Roycroft Press |
1524 |

Art is the expression of man's joy in his work. | |

Elbert Hubbard, from 101 Epigrams According to Elbert Hubbard, Roycroft Press |
1523 |

Art is not a thing: it is a way. | |

Elbert Hubbard, from 101 Epigrams According to Elbert Hubbard, Roycroft Press |
1522 |

Art is not a thing, it is a way. | |

Elbert Hubbard, |
1235 |

One machine can do the work of fifty ordinary men. No machine can do the work of one extraordinary man. | |

Elbert Hubbard, |
831 |

You should really think of my pictures [of fractals] as a metaphor for all living things. | |

John Hubbard, from the videotape "The Beauty and Complexity of the Mandelbrot Set". |
481 |

An Idea that is not dangerous is unworthy of being called an idea at all. | |

Elbert Hubbard, from 101 Epigrams According to Elbert Hubbard, Roycroft Press |
1521 |

What happens to a dream deferred? Does it dry up- Like a raisin in the sun? Or fester like a sore- And then run? Does it stink like rotten meat? Or crust and sugar over- Like a syrupy sweet? Maybe it just sags- Like a heavy load. Or does it explode? | |

Langston Hughes, (A Dream Deferred). |
178 |

Oh God of dust and rainbows, help us see that without dust the rainbow would not be. | |

Langston Hughes, |
1442 |

By this means all knowledge degenerates into probability; … There is Algebraist nor Mathematicians so expert in his science, as to place entire confidence in any truth immediately upon his discovery of it, or regard it as any thing, but a mere probability. Every time her runs over his proofs, his confidence encreases; but still more by the approbation of his friends; and is raised to its utmost perfection by the universal assent and applauses of the learned world. Now 'tis evident that this gradual encrease of assurance is nothing but the addition of new probabilities. | |

David Hume, from Treatise on Human Nature, quoted in What is Mathematics, Really? by Rueben Hersch, p. 189. |
1254 |

In 1953 I realized that the straight line leads to the downfall of mankind. But the straight line has become an absolute tyranny. The straight line is something cowardly drawn with a rule, without thought or feeling; it is the line which does not exist in nature... Any design undertaken with the straight line will be stillborn. Today we are witnessing the triumph of rationalist knowhow and yet, at the same time, we find ourselves confronted with emptiness. An esthetic void, desert of uniformity, criminal sterility, loss of creative power. Even creativity is prefabricated. We have become impotent. We are no longer able to create. That is our real illiteracy. | |

Friedensreich Hundertwasser, quoted in The Beauty of Fractals, by H.O. Peitgen and P.H. Richter. |
478 |

Research is formalized curiosity. | |

Zora Neale Hurston, |
1023 |

The method of fluxions is probably one the greatest, most subtle, and sublime discoveries of any age; it opens a new world to our view, and extends our knowledge, as it were, to infinity; carrying us beyond the bounds that seemed to have been prescribed to the human mind, at least infinitely beyond those to which the ancient geometry was confined. | |

Charles Hutton, quoted in Memorabilia Mathematica, by R. Moritz. |
179 |

This universe, I conceive, like to a great game being played out, and we poor mortals are allowed to take a hand. By great good fortune the wiser among us have made out some few of the rules of the game, as at present played. We call them 'Laws of Nature', and honor them because we find that if we obey them we win something for our pains. The cards are our theories and hypotheses, the tricks our experimental verifications. But what sane man would endeavor to solve this problem? …The problem of the metaphysicians is to my mind no saner. | |

Thomas Huxley, quoted on p. 513 of The Colossal Book of Mathematics by Martin Gardner |
1707 |

Mathematics may be compared to a mill of exquisite workmanship, which grinds you stuff of any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat-flour from peascods, so pages of formulae will not get a definite result out of loose data. Mathematics may be compared to a mill of exquisite workmanship, which grinds you stuff of any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat-flour from peascods, so pages of formulae will not get a definite result out of loose data. | |

Thomas Huxley, |
1594 |

I warn you I will stop at no point so long as clear reasoning will take me further. | |

Thomas Huxley, |
1593 |

The known is finite, the unknown infinite; intellectually we stand on an islet in the midst of an illimitable ocean of inexplicability. Our business in every generation is to reclaim a little more land. | |

Thomas Huxley, |
1591 |

The chess-board is the world; the pieces are the phenomena of the universe; the rules of the game are what we call the laws of Nature. The player on the other side is hidden from us. We know that his play is always fair, just and patient. But we know, to our cost, that he never overlooks a mistake, or makes the slightest allowance for ignorance. | |

Thomas Huxley, |
1590 |

That men do not learn very much from the lessons of history is the most important of all the lessons that history has to teach. | |

Aldous Huxley, from "See no evil, hear no evil, speak no evil" in Notices of the American Mathematical Society, vol. 45, no. 9, October 1998. |
744 |

The rung of a ladder was never meant to rest upon, but only to hold a man's foot long enough to enable him to put the other somewhat higher. | |

Thomas Huxley, quoted Essentials of Mathematics by Margie Hale. |
955 |

The greatest tragedy of science - the slaying of a beautiful hypothesis by an ugly fact. | |

Thomas Huxley, |
180 |

Sooner or later, false thinking brings wrong conduct. | |

Julian Huxley, |
561 |

If we evolved a race of Isaac Newtons, that would not be progress. For the price Newton had to pay for being a supreme intellect was that he was incapable of friendship, love, fatherhood, and many other desirable things. As a man he was a failure; as a monster he was superb. | |

Aldous Huxley, from Interview with J. W. N. Sullivan, Contemporary Mind, London, 1934 |
860 |

At least two thirds of our miseries spring from human stupidity, human malice and those great motivators and justifiers of malice and stupidity, idealism, dogmatism and proselytizing zeal on behalf of religious or political idols. | |

Aldous Huxley, |
1379 |

…You [Leibniz] will not deny that you have discovered a very remarkable property of the circle [pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...], which will forever be famous among geometers. | |

Huygens, quoted in Analysis by Its History by E. Hairer and G. Wanner. |
1102 |

147 quotes found and displayed.