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Results Quotes

“Mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It's the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.”

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Author:Ivars Peterson
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“Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning. The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings.”

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Author:Alan Turing
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“Toward the end of his life, Gödel feared that he was being poisoned, and he starved himself to death. His theorem is one of the most extraordinary results in mathematics, or in any intellectual field in this century. If ever potential mental instability is detectable by genetic analysis, an embryo of someone with Kurt Gödel's gifts might be aborted.”

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Book:The Ascent of Science
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“The page of my notebook was filled with many messy integrals, but all of a sudden I saw emerge a formula for counting. I had begun to calculate a quantity on the assumption that the result was a real number, but found instead that, in certain units, all the possible answers would be integers. This meant that areas and volumes cannot take any value, but come in multiples of fixed units.”

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Book:Three Roads To Quantum Gravity
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“Logic, too, also rests on assumptions that do not correspond to anything in the real world, e.g., on the assumption that there areequal things, that the same thing is identical at different points in time: but this science arose as a result of the opposite belief (that such things actually exist in the real world). And it is the same with mathematics, which would certainly never have arisen if it had been understood from the beginning that there is no such thing in nature as a perfectly straight line, a true circle, and absolute measure.”

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Author:Friedrich Nietzsche
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“There is plenty of room left for exact experiment in art, and the gate has been opened for some time. What had been accomplished in music by the end of the eighteenth century has only begun in the fine arts. Mathematics and physics have given us a clue in the form of rules to be strictly observed or departed from, as the case may be. Here salutary discipline is come to grips first of all with the function of forms, and not with form as the final result … in this way we learn how to look beyond the surface and get to the root of things.”

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Author:Paul Klee
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“However the machine would permit us to test the hypothesis for any special value of n. We could carry out such tests for a sequence of consecutive values n=2,3,.. up to, say, n=100. If the result of at least one test were negative, the hypothesis would prove to be false; otherwise our confidence in the hypothesis would increase, and we should feel encouraged to attempt establishing the hypothesis, instead of trying to construct a counterexample.”

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Author:Alfred Tarski
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“Too much knowledge could be a bad thing. I was lead to the Szemerédi theorem by proving a result, about squares, that Euler had already proven, and I relied on an "obvious" fact, about arithmetical progressions, that was unproved at the time. But that lead me to try and prove that formerly unproved statement- about arithmetical progressions-and that ultimately lead to the Szemerédi Theorem.”

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Author:Endre Szemeredi
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“There are fields of scientific work...which have been explored from the different sides of pure mathematics, statistics, electrical engineering, and neurophysiology...in which every single notion receives a separate and different name from each group, and in which important work has been triplicated or quadruplicated, while still other important work is delayed by the unavailability in one field of results that may have already become classical in the next field.”

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Author:Norbert Wiener
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