Citas de números enteros

• The trouble with integers is that we have examined only the very small ones. Maybe all the exciting stuff happens at really big numbers, ones we can't even begin to think about in any very definite way. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions.

  Ronald Graham

  Rain, Math, Thinking

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• Los números enteros son la fuente de todas las matemáticas.

  Hermann Minkowski

  Fountainhead, Mathematics

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• The series of integers is obviously an invention of the human mind, a self-created tool which simplifies the ordering of certain sensory experiences.

  Albert Einstein

  Education, Self, Mind

  Albert Einstein (2010). “Ideas And Opinions”, p.22, Broadway Books

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• Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next.

  Gregory Bateson

  Mean, Numbers, Next

  "El número es diferente de la cantidad". CoEvolution Quarterly, www.oikos.org. Primavera de 1978.

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• El Buen Dios hizo todos los números enteros; el resto es obra del hombre.

  Leopold Kronecker

  Math, Men, Lord

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• Nature does not count nor do integers occur in nature. Man made them all, integers and all the rest, Kronecker to the contrary notwithstanding.

  Percy Williams Bridgman

  Math, Men, Doe

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• All of mathematics can be deduced from the sole notion of an integer; here we have a fact universally acknowledged today.

  Emile Borel

  Numbers, Today, Facts

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• The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules... One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms.

  Kurt Gödel

  Simple, Two, Development

  "Kurt Gödel: Collected Works: Volume I: Publications 1929-1936".

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• My problem is that I have been persecuted by an integer.

  George Armitage Miller

  Problem, Persecuted, Has Beens

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• Dios hizo los números enteros, el hombre hizo el resto.

  Leopold Kronecker

  Men, Made

  Quoted in F Cajori A History of Mathematics (1919).

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• If we can dispel the delusion that learning about computers should be an activity of fiddling with array indexes and worrying whether X is an integer or a real number, we can begin to focus on programming as a source of ideas.

  Hal Abelson

  Real, Ideas, Numbers

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• Is a woman a thinking unit at all, or a fraction always wanting its integer?

  Thomas Hardy

  Thinking, Fractions, Units

  Thomas Hardy (2016). “Jude the Obscure: Works of Hardy”, p.314, 谷月社

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• [L]ife ceases to be a fraction and becomes an integer.

  Harry Emerson Fosdick

  Fractions, Cease

  Harry Emerson Fosdick (1971). "Los problemas reales de la gente real: Soluciones para los cristianos"

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• En esta comunicación deseo mostrar, en primer lugar, en el caso más simple del átomo de hidrógeno (no relativista y no distorsionado), que los tipos habituales para la cuantización pueden sustituirse por otro requisito, en el que ya no se mencionan los "números enteros". En su lugar, los números enteros aparecen de la misma forma natural que los números enteros que especifican el número de nodos de una cuerda vibrante. La nueva concepción puede generalizarse, y creo que toca el significado más profundo de las reglas cuánticas.

  Erwin Schrodinger

  Creer, Comunicación, Ciencia

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• Todos los resultados de la investigación matemática más profunda deben ser expresables en última instancia en la forma simple de propiedades de los números enteros.

  Leopold Kronecker

  Simple, Mathematics, Form

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• I read in the proof sheets of Hardy on Ramanujan: "As someone said, each of the positive integers was one of his personal friends." My reaction was, "I wonder who said that; I wish I had." In the next proof-sheets I read (what now stands), "It was Littlewood who said..."

  John Edensor Littlewood

  Friendship, Math, Wish

  John Edensor Littlewood, Béla Bollobás (1986). “Littlewood's Miscellany”, p.61, Cambridge University Press

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• Los números son el producto de contar. Las cantidades son el producto de la medición. Esto significa que los números pueden ser exactos porque hay una discontinuidad entre cada número entero y el siguiente. Entre dos y tres hay un salto. En el caso de la cantidad, no existe tal salto; y como el salto falta en el mundo de la cantidad, es imposible que ninguna cantidad sea exacta. Se pueden tener exactamente tres tomates. Nunca puedes tener exactamente tres litros de agua. Siempre la cantidad es aproximada.

  Gregory Bateson

  Media, Dos, Números

  "El número es diferente de la cantidad". CoEvolution Quarterly, www.oikos.org. Primavera de 1978.

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• El desarrollo de las matemáticas hacia una mayor precisión ha conducido, como es bien sabido, a la formalización de grandes extensiones de las mismas, de modo que se puede demostrar cualquier teorema utilizando únicamente unas pocas reglas mecánicas.

  Kurt Gödel

  Desarrollo, Conocidas, Matemáticas

  Kurt Gödel, S. Feferman (1986). "Kurt Gödel: Obras Completas: Volume I: Publications 1929-1936", p.145, Oxford University Press

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• The gods and their tranquil abodes appear, which no winds disturb, nor clouds bedew with showers, nor does the white snow, hardened by frost, annoy them; the heaven, always pure, is without clouds, and smiles with pleasant light diffused.

  Lucrecio

  Clouds, Light, Wind

  "De Rerum Natura". Book by Lucretius, III. 18,

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• Cada vez se presiona más a los estudiantes para que se sometan a más exámenes, reduciéndolos a filas y filas de números enteros negros sobre un expediente académico, listos para pasar directamente a un ordenador.

  Leah Hager Cohen

  Black, Tests, Pressure

  Leah Hager Cohen (2003). “Heart, You Bully, You Punk”, Viking Press

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• By itself, an ordinary snapshot is no less banal than the petite madeleine in Proust's In Search of Lost Time... but as goad to memory, it is often the first integer in a sequence of recollections that has the power to deny time for the sake of love.

  Michael Lesy

  Memories, Sake, Firsts

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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.

  Lee Smolin

  Notebook, Real, Numbers

  Lee Smolin (2008). “Three Roads To Quantum Gravity”, p.106, Hachette UK

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• I have tried, with little success, to get some of my friends to understand my amazement that the abstraction of integers for counting is both possible and useful. Is it not remarkable that 6 sheep plus 7 sheep makes 13 sheep; that 6 stones plus 7 stones make 13 stones? Is it not a miracle that the universe is so constructed that such a simple abstraction as a number is possible? To me this is one of the strongest examples of the unreasonable effectiveness of mathematics. Indeed, I find it both strange and unexplainable.

  Richard Hamming

  Simple, Effectiveness, Numbers

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• La aritmética comienza con los números enteros y procede ampliando sucesivamente el sistema numérico con números racionales y negativos, números irracionales, etc... Pero el siguiente paso bastante lógico después de los reales, a saber, la introducción de los infinitesimales, simplemente se ha omitido. Creo que en los siglos venideros se considerará una gran rareza en la historia de las matemáticas que la primera teoría exacta de los infinitesimales se desarrollara 300 años después de la invención del cálculo diferencial.

  Abraham Robinson

  Pensar, rarezas, descubrimiento

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