(Last Updated On: December 13, 2017)
Hadamard, Jacques The shortest path between two truths in the real domain passes through the complex domain.
Quoted in The Mathematical Intelligencer, v. 13, no. 1, Winter 1991.
Hadmard, Jacques Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Haldane, John Burdon Sanderson (1892-1964) In scientific thought we adopt the simplest theory which will explain all the facts under consideration and enable us to predict new facts of the same kind. The catch in this criterion lies in the world “simplest.” It is really an aesthetic canon such as we find implicit in our criticisms of poetry or painting. The layman finds such a law as dx/dt = K(d^2x/dy^2) much less simple than “it oozes,” of which it is the mathematical statement. The physicist reverses this judgment, and his statement is certainly the more fruitful of the two, so far as prediction is concerned. It is, however, a statement about something very unfamiliar to the plainman, namely, the rate of change of a rate of change.
Possible Worlds, 1927.
Haldane, John Burdon Sanderson (1892-1964) A time will however come (as I believe) when physiology will invade and destroy mathematical physics, as the latter has destroyed geometry.
Daedalus, or Science and the Future, London: Kegan Paul, 1923.
Halmos, Paul R. 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.
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.
Halmos, Paul R. … the student skit at Christmas contained a plaintive line: “Give us Master’s exams that our faculty can pass, or give us a faculty that can pass our Master’s exams.”
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.
Halmos, Paul R. I remember one occasion when I tried to add a little seasoning to a review, but I wasn’t allowed to. The paper was by Dorothy Maharam, and it was a perfectly sound contribution to abstract measure theory. The domains of the underlying measures were not sets but elements of more general Boolean algebras, and their range consisted not of positive numbers but of certain abstract equivalence classes. My proposed first sentence was: “The author discusses valueless measures in pointless spaces.”
I want to be a Mathematician, Washington: MAA Spectrum, 1985, p. 120.
Halmos, Paul R. …the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.
Halmos, Paul R. The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me — both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tension.
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.
Halmos, Paul R. Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.
Halmos, Paul R. To be a scholar of mathematics you must be born with talent, insight, concentration, taste, luck, drive and the ability to visualize and guess.
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.
Hamilton, [Sir] William Rowan (1805-1865) Who would not rather have the fame of Archimedes than that of his conqueror Marcellus?
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber and Schmidt, 1971.
Hamilton, Sir William Rowan (1805-1865) I regard it as an inelegance, or imperfection, in quaternions, or rather in the state to which it has been hitherto unfolded, whenever it becomes or seems to become necessary to have recourse to x, y, z, etc..
In a letter from Tait to Cayley.
Hamilton, Sir William Rowan (1805-1865) On earth there is nothing great but man; in man there is nothing great but mind.
Lectures on Metaphysics.
Hamming, Richard W. Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Hamming, Richard W. Mathematics is an interesting intellectual sport but it should not be allowed to stand in the way of obtaining sensible information about physical processes.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Hardy, Godfrey H. (1877 – 1947) [On Ramanujan]
I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”
Ramanujan, London: Cambridge Univesity Press, 1940.
Hardy, Godfrey H. (1877 – 1947) Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.
A Mathematician’s Apology, London, Cambridge University Press, 1941.
Hardy, Godfrey H. (1877 – 1947) I am interested in mathematics only as a creative art.
A Mathematician’s Apology, London, Cambridge University Press, 1941.
Hardy, Godfrey H. (1877 – 1947) Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.
Hardy, Godfrey H. (1877 – 1947) In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy.
A Mathematician’s Apology, London, Cambridge University Press, 1941.
Hardy, Godfrey H. (1877 – 1947) 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.
A Mathematician’s Apology, London, Cambridge University Press, 1941.
Hardy, Godfrey H. (1877 – 1947) Young Men should prove theorems, old men should write books.
Quoted by Freeman Dyson in Freeman Dyson: Mathematician, Physicist, and Writer. Interview with Donald J. Albers, The College Mathematics Journal, vol. 25, No. 1, January 1994.
Hardy, Godfrey H. (1877 – 1947) A science is said to be useful of its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life.
A Mathematician’s Apology, London, Cambridge University Press, 1941.
Hardy, Godfrey H. (1877 – 1947) The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.
A Mathematician’s Apology, London, Cambridge University Press, 1941.
Hardy, Godfrey H. (1877 – 1947) I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our “creations,” are simply the notes of our observations.
A Mathematician’s Apology, London, Cambridge University Press, 1941.
Hardy, Godfrey H. (1877 – 1947) Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean.
A Mathematician’s Apology, London, Cambridge University Press,1941.
Hardy, Godfrey H. (1877 – 1947) 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.
A Mathematician’s Apology, London, Cambridge University Press, 1941.
Hardy, Thomas …he seemed to approach the grave as an hyperbolic curve approaches a line, less directly as he got nearer, till it was doubtful if he would ever reach it at all.
Harish-Chandra I have often pondered over the roles of knowledge or experience, on the one hand, and imagination or intuition, on the other, in the process of discovery. I believe that there is a certain fundamental conflict between the two, and knowledge, by advocating caution, tends to inhibit the flight of imagination. Therefore, a certain naivete, unburdened by conventional wisdom, can sometimes be a positive asset.
R. Langlands, “Harish-Chandra,” Biographical Memoirs of Fellows of the Royal Society 31 (1985) 197 – 225.
Harris, Sydney J. The real danger is not that computers will begin to think like men, but that men will begin to think like computers.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Hawking, Stephen Williams (1942- ) God not only plays dice. He also sometimes throws the dice where they cannot be seen.
[See related quotation from Albert Einstein.] Nature 1975 257.
Heath, Sir Thomas [The works of Archimedes] are without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern elimination of everything not immediately relevant to the purpose, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.
A History of Greek Mathematics. 1921.
Heaviside, Oliver (1850-1925) [Criticized for using formal mathematical manipulations, without understanding how they worked:]
Should I refuse a good dinner simply because I do not understand the process of digestion?
Heinlein, Robert A. Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house.
Time Enough for Love.
Heisenberg, Werner (1901-1976) An expert is someone who knows some of the worst mistakes that can be made in his subject, and how to avoid them.
Physics and Beyond. 1971.
Hempel, Carl G. The propositions of mathematics have, therefore, the same unquestionable certainty which is typical of such propositions as “All bachelors are unmarried,” but they also share the complete lack of empirical content which is associated with that certainty: The propositions of mathematics are devoid of all factual content; they convey no information whatever on any empirical subject matter.
“On the Nature of Mathematical Truth” in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Hempel, Carl G. The most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which accounts for its fame as the queen of the sciences, is no doubt the peculiar certainty and necessity of its results.
“Geometry and Empirical Science” in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Hempel, Carl G. …to characterize the import of pure geometry, we might use the standard form of a movie-disclaimer: No portrayal of the characteristics of geometrical figures or of the spatial properties of relationships of actual bodies is intended, and any similarities between the primitive concepts and their customary geometrical connotations are purely coincidental.
“Geometry and Empirical Science” in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Henkin, Leon One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed. This gives students the idea that there is a book somewhere with all the right answers to all of the interesting questions, and that teachers know those answers. And if one could get hold of the book, one would have everything settled. That’s so unlike the true nature of mathematics.
L.A. Steen and D.J. Albers (eds.), Teaching Teachers, Teaching Students, Boston: Birkhäuser, 1981, p89.
Hermite, Charles (1822 – 1901) There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation.
In The Mathematical Intelligencer, v. 5, no. 4.
Hermite, Charles (1822-1901) Abel has left mathematicians enough to keep them busy for 500 years.
In G. F. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Hermite, Charles (1822-1901) We are servants rather than masters in mathematics.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Hertz, Heinrich One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser that we are, wiser even than their discoverers, that we get more out of them than was originally put into them.
Quoted by ET Bell in Men of Mathematics, New York, 937.
Hesse, Hermann (1877-1962) You treat world history as a mathematician does mathematics, in which nothing but laws and formulae exist, no reality, no good and evil, no time, no yesterday, no tomorrow, nothing but an eternal, shallow, mathematical present.
Hilbert, David (1862-1943) Wir müssen wissen.
Wir werden wissen.
[Engraved on his tombstone in Göttingen.]
Hilbert, David (1862-1943) Before beginning I should put in three years of intensive study, and I haven’t that much time to squander on a probable failure.
[On why he didn’t try to solve Fermat’s last theorem]
Quoted in E.T. Bell Mathematics, Queen and Servant of Science, New York: McGraw Hill Inc., 1951.
Hilbert, David (1862-1943) Galileo was no idiot. Only an idiot could believe that science requires martyrdom – that may be necessary in religion, but in time a scientific result will establish itself.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1971.
Hilbert, David (1862-1943) Mathematics is a game played according to certain simple rules with meaningless marks on paper.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Hilbert, David (1862-1943) Physics is much too hard for physicists.
C. Reid Hilbert, London: Allen and Unwin, 1970.
Hilbert, David (1862-1943) How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments.
Hilbert, David (1862-1943) The art of doing mathematics consists in finding that special case which contains all the germs of generality.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Hilbert, David (1862-1943) The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Hilbert, David (1862-1943)
I have tried to avoid long numerical computations, thereby following Riemann’s postulate that proofs should be given through ideas and not voluminous computations.
Report on Number Theory, 1897.
Hilbert, David (1862-1943) One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it.
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber and Schmidt,1971.
Hilbert, David (1862-1943) Mathematics knows no races or geographic boundaries; for mathematics,the cultural world is one country.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Hilbert, David (1862-1943) The infinite! No other question has ever moved so profoundly the spirit of man.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Hirst, Thomas Archer 10th August 1851: On Tuesday evening at Museum, at a ball in the gardens. The night was chill, I dropped too suddenly from Differential Calculus into ladies’ society, and could not give myself freely to the change. After an hour’s attempt so to do, I returned, cursing the mode of life I was pursuing; next morning I had already shaken hands, however, with Diff. Calculus, and forgot the ladies….
J. Helen Gardner and Robin J. Wilson, “Thomas Archer Hirst – Mathematician Xtravagant II – Student Days in Germany”, The American Mathematical Monthly , v. 6, no. 100.
Hobbes, Thomas There is more in Mersenne than in all the universities together.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Hobbes, Thomas To understand this for sense it is not required that a man should be a geometrician or a logician, but that he should be mad.
[“This” is that the volume generated by revolving the region under 1/x from 1 to infinity has finite volume.]
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Hobbes, Thomas Geometry, which is the only science that it hath pleased God hitherto to bestow on mankind.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Hobbes, Thomas The errors of definitions multiply themselves according as the reckoning proceeds; and lead men into absurdities, which at last they see but cannot avoid, without reckoning anew from the beginning.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Holmes, Oliver Wendell Descartes commanded the future from his study more than Napoleon from the throne.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Holmes, Oliver Wendell Certitude is not the test of certainty. We have been cocksure of many things that are not so.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Holmes, Oliver Wendell I was just going to say, when I was interrupted, that one of the many ways of classifying minds is under the heads of arithmetical and algebraical intellects. All economical and practical wisdom is an extension of the following arithmetical formula: 2 + 2 = 4. Every philosophical proposition has the more general character of the expression a + b = c. We are mere operatives, empirics, and egotists until we learn to think in letters instead of figures.
The Autocrat of the Breakfast Table.
Holt, M. and Marjoram, D. T. E. The truth of the matter is that, though mathematics truth may be beauty, it can be only glimpsed after much hard thinking. Mathematics is difficult for many human minds to grasp because of its hierarchical structure: one thing builds on another and depends on it.
Mathematics in a Changing World Walker, New York 1973.
Hofstadter, Douglas R. (1945 – ) Hofstadter’s Law: It always takes longer than you expect, even when you take into account Hofstadter’s Law.
Gödel, Escher, Bach 1979.
Hughes, Richard Science, being human enquiry, can hear no answer except an answer couched somehow in human tones. Primitive man stood in the mountains and shouted against a cliff; the echo brought back his own voice, and he believed in a disembodied spirit. The scientist of today stands counting out loud in the face of the unknown. Numbers come back to him – and he believes in the Great Mathematician.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Hume, David (1711 – 1776) If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, `Does it contain any abstract reasoning concerning quantity or number?’ No. `Does it contain any experimental reasoning concerning matter of fact and existence?’ No. Commit it then to the flames: for it can contain nothing but sophistry and illusion.
Treatise Concerning Human Understanding.
Huxley, Aldous I admit that mathematical science is a good thing. But excessive devotion to it is a bad thing.
Interview with J. W. N. Sullivan, Contemporary Mind, London, 1934.
Huxley, Aldous 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.
Interview with J. W. N. Sullivan, Contemporary Mind, London, 1934.
Huxley, Aldous …[he] was as much enchanted by the rudiments of algebra as he would have been if I had given him an engine worked by steam, with a methylated spirit lamp to heat the boiler; more enchanted, perhapsfor the engine would have got broken, and, remaining always itself, would in any case have lost its charm, while the rudiments of algebra continued to grow and blossom in his mind with an unfailing luxuriance. Every day he made the discovery of something which seemed to him exquisitely beautiful; the new toy was inexhaustible in its potentialities.
Young Archimedes.
Huxley, Thomas Henry (1825-1895) This seems to be one of the many cases in which the admitted accuracy of mathematical processes is allowed to throw a wholly inadmissible appearance of authority over the results obtained by them. Mathematics may be compared to a mill of exquisite workmanship, which grinds your stuff of any degree of fineness; but, nevertheless, what you get out depends on 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.
Quarterly Journal of the Geological Society, 25,1869.
Huxley, Thomas Henry (1825-1895) The mathematician starts with a few propositions, the proof of which is so obvious that they are called selfevident, and the rest of his work consists of subtle deductions from them. The teaching of languages, at any rate as ordinarily practised, is of the same general nature authority and tradition furnish the data, and the mental operations are deductive.
“Scientific Education -Notes of an After-dinner Speech.” Macmillan’s Magazine Vol XX, 1869.
Huxley, Thomas Henry (1825-1895) It is the first duty of a hypothesis to be intelligible.

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