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Goethe Mathematics has the completely false reputation of yielding infallible conclusions. Its infallibility is nothing but identity. Two times two is not four, but it is just two times two, and that is what we call four for short. But four is nothing new at all. And thus it goes on and on in its conclusions, except that in the higher formulas the identity fades out of sight. In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956, p. 1754.
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Goodman, Nicholas P. There are no deep theorems — only theorems that we have not understood very well. The Mathematical Intelligencer, vol. 5, no. 3, 1983.
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Gordon, P This is not mathematics, it is theology. [On being exposed to Hilbert’s work in invariant theory.] Quoted in P. Davis and R. Hersh The Mathematical Experience, Boston: Birkh’ser, 1981.
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Graham, Ronald It wouild be very discouraging if somewhere down the line you could ask a computer if the Riemann hypothesis is correct and it said, `Yes, it is true, but you won’t be able to understand the proof.’ John Horgan. Scientific American 269:4 (October 1993) 92-103.
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Grnbaum, Branko (1926 – ), and Shephard, G. C. (?) Mathematicians have long since regarded it as demeaning to work on problems related to elementary geometry in two or three dimensions, in spite of the fact that it it precisely this sort of mathematics which is of practical value. Handbook of Applicable Mathematics.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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. Far from the Madding Crowd.
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