Frases de Gottlob Frege (29 citas)

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Los fundamentos de la aritmética

Gottlob Frege

Gottlob Frege: 29 frases 4 Me gusta

Frases célebres de Gottlob Frege

“Todo buen matemático es, al menos, la mitad de un filósofo, y todo buen filósofo es, al menos, la mitad de un matemático.”

Gottlob Frege

Fuente: Atribuido a Frege en: A.A.B. Aspeitia (2000) Mathematics as grammar: 'Grammar' in Wittgenstein's philosophy of mathematics during the Middle Period. Universidad de Indiana. pág. 25

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“A menudo, solo después de un inmenso esfuerzo intelectual, que puede haber continuado durante siglos, la humanidad finalmente logra alcanzar el conocimiento de un concepto en su forma pura, quitando las acumulaciones irrelevantes que lo ocultan del ojo de la mente.”

Gottlob Frege

“Espero poder demostrar en el presente trabajo que es probable que las leyes de la aritmética sean juicios analíticos y, en consecuencia, a priori. La Aritmética se convierte así en un simple desarrollo de la Lógica, y toda proposición aritmética procede de una ley lógica, aunque sea derivada de alguna. Aplicar la aritmética en las ciencias físicas es llevar la lógica para que soporte los hechos observados; el cálculo es deducción.”

Gottlob Frege libro Los fundamentos de la aritmética

Fuente: Frege (1884) The Foundations of Arithmetic, pág. 99.

“Cada enunciado que tenemos por verdadero, es conocido o bien por medio de la experiencia o en razón de su significado. No hay más fuentes de conocimiento que el dato de los sentidos o el significado que le damos al enunciado.”

Gottlob Frege

Fuente: Lewis, C. I. The modes of meaning. pág 15-16 ‘Every statement we know to be true is so known either by reason of experience or by reason of what the statement itself means. There are no other sources of knowledge than on the one hand data of sense and on the other hand our own intended meanings.’

“El realismo, por tanto, es el punto de vista que sostiene que la matemática es la ciencia de los números, conjuntos, funciones, etc., tal y como la física es el estudio de los objetos físicos ordinarios, cuerpos astronómicos y partículas subatómicas entre otros. Esto es, la matemática trata acerca de esos objetos, y es el modo en que tales objetos son lo que hace a los enunciados de la matemática verdaderos o falsos.”

Gottlob Frege

Fuente: Maddy, Penélope. Realism in mathematics. Pg. 2 ‘Realism, then, is the view that mathematics is the science of numbers, sets, functions, etc., just as physical science is the study of ordinary physical objects, astronomical bodies, subatomic particles, and so on. That is, mathematics is about these things, and the way these things are is what makes mathematical statements true or false.’

“El matemático está interesado, por lo tanto, bajo este enfoque, en la descripción correcta de un ámbito particular de la realidad, comparable con los reinos físicos descritos por el geógrafo y el astrónomo.”

Gottlob Frege

Fuente: Dummett, Michael. El Platonismo, en la verdad y otros enigmas. FCE. 1990 pág. 282.

“Comparo la Aritmética con un árbol que se desarrolla hacia arriba en una multitud de técnicas y teoremas mientras que la raíz se adentra en las profundidades.”

Gottlob Frege

Fuente: Grundgesetze der Arithmetik (1893), xiii,

Gottlob Frege: Frases en inglés

“Often it is only after immense intellectual effort, which may have continued over centuries, that humanity at last succeeds in achieving knowledge of a concept in its pure form, by stripping off the irrelevant accretions which veil it from the eye of the mind.”

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Gottlob Frege

Translation J. L. Austin (Oxford, 1950) as quoted by Stephen Toulmin, Human Understanding: The Collective Use and Evolution of Concepts (1972) Vol. 1, p. 56.
Grundgesetze der Arithmetik, 1893 and 1903

“If the task of philosophy is to break the domination of words over the human mind […], then my concept notation, being developed for these purposes, can be a useful instrument for philosophers […] I believe the cause of logic has been advanced already by the invention of this concept notation.”

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Gottlob Frege libro Begriffsschrift

Begriffsschrift (1879) Preface to the Begriffsschrift

“Being true is different from being taken as true, whether by one or by many or everybody, and in no case is it to be reduced to it. There is no contradiction in something's being true which everybody takes to be false. I understand by 'laws of logic' not psychological laws of takings-to-be-true, but laws of truth. …If being true is thus independent of being acknowledged by somebody or other, then the laws of truth are not psychological laws: they are boundary stones set in an eternal foundation, which our thought can overflow, but never displace. It is because of this that they have authority for our thought if it would attain truth. They do not bear the relation to thought that the laws of grammar bear to language; they do not make explicit the nature of our human thinking and change as it changes.”

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Gottlob Frege

Introduction, Tr. Montgomery Furth (1964)
Grundgesetze der Arithmetik, 1893 and 1903

“Without some affinity in human ideas art would certainly be impossible; but it can never be exactly determined how far the intentions of the poet are realized.”

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Gottlob Frege Sense and reference

Gottlob Frege (1892). On Sense and Reference.
Über Sinn und Bedeutung, 1892

“It really is worth the trouble to invent a new symbol if we can thus remove not a few logical difficulties and ensure the rigour of the proofs. But many mathematicians seem to have so little feeling for logical purity and accuracy that they will use a word to mean three or four different things, sooner than make the frightful decision to invent a new word.”

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Gottlob Frege

Gottlob Frege in: Dagobert David Runes (1962). Readings in epistemology, theory of knowledge and dialectics. p. 334

“Equality gives rise to challenging questions which are not altogether easy to answer… a = a and a = b are obviously statements of differing cognitive value; a = a holds a priori and, according to Kant, is to be labeled analytic, while statements of the form a = b often contain very valuable extensions of our knowledge and cannot always be established a priori. The discovery that the rising sun is not new every morning, but always the same, was one of the most fertile astronomical discoveries. Even to-day the identification of a small planet or a comet is not always a matter of course. Now if we were to regard equality as a relation between that which the names 'a' and 'b' designate, it would seem that a = b could not differ from a = a (i. e. provided a = b is true). A relation would thereby be expressed of a thing to itself, and indeed one in which each thing stands to itself but to no other thing.”

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Gottlob Frege Sense and reference

As cited in: M. Fitting, Richard L. Mendelsoh (1999), First-Order Modal Logic, p. 142. They called this Frege's Puzzle.
Über Sinn und Bedeutung, 1892

“A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.”

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Gottlob Frege

Note in the appendix of Grundlagen der Arithmetik (Vol. 2) after Frege had received a letter of Bertrand Russell in which Russell had explained his discovery of, what is now known as, Russell's paradox.
Grundgesetze der Arithmetik, 1893 and 1903

“If I compare arithmetic with a tree that unfolds upward into a multitude of techniques and theorems while its root drives into the depths, then it seems to me that the impetus of the root.”

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Gottlob Frege

Gottlob Frege, Montgomery Furth (1964). The Basic Laws of Arithmetic: Exposition of the System. p. 10

“Is it always permissible to speak of the extension of a concept, of a class? And if not, how do we recognize the exceptional cases? Can we always infer from the extension of one concept's coinciding with that of a second, that every object which falls under the first concept also falls under the second?”

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Gottlob Frege

Vol. 2, p. 127. Replying to Bertrand Russell's letter about Russell's Paradox; quoted in The Stanford Encyclopedia of Philosophy http://plato.stanford.edu/entries/russell-paradox/
Grundgesetze der Arithmetik, 1893 and 1903

“The ideal of strictly scientific method in mathematics which I have tried to realise here, and which perhaps might be named after Euclid I should like to describe in the following way… The novelty of this book does not lie in the content of the theorems but in the development of the proofs and the foundations on which they are based… With this book I accomplish an object which I had in view in my Begriffsschrift of 1879 and which I announced in my Grundlagen der Arithmetik. I am here trying to prove the opinion on the concept of number that I expressed in the book last mentioned.”

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Gottlob Frege

Vol. 1. pp. 137-140, as cited in: Ralph H. Johnson (2012), Manifest Rationality: A Pragmatic Theory of Argument, p. 87
Grundgesetze der Arithmetik, 1893 and 1903

“Facts, facts, facts,' cries the scientist if he wants to emphasize the necessity of a firm foundation for science. What is a fact? A fact is a thought that is true. But the scientist will surely not recognize something which depends on men's varying states of mind to be the firm foundation of science.”

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Gottlob Frege

Gottlob Frege (1956). "The thought: A logical inquiry" in: Peter Ludlow (1997) Readings in the Philosophy of Language. p. 27

“A judgment, for me is not the mere grasping of a thought, but the admission of its truth.”

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Gottlob Frege Sense and reference

Gottlob Frege (1892). On Sense and Reference, note 7.
Über Sinn und Bedeutung, 1892

“This ideography is a "formula language", that is, a lingua characterica, a language written with special symbols, "for pure thought", that is, free from rhetorical embellishments, "modeled upon that of arithmetic", that is, constructed from specific symbols that are manipulated according to definite rules.”

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Gottlob Frege

paraphrasing Frege's Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought (1879) in Jean Van Heijenoort ed., in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (1967)

“We suppose, it would seem, that concepts grow in the individual mind like leaves on a tree, and we think to discover their nature by studying their growth; we seek to define them psychologically, in terms of the human mind. But this account makes everything subjective, and if we follow it through to the end, does away with truth. What is known as the history of concepts is really a history either of our knowledge of concepts or of the meanings of words.”

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Gottlob Frege

“The historical approach, with its aim of detecting how things began and arriving from these origins at a knowledge of their nature, is certainly perfectly legitimate; but it also has its limitations. If everything were in continual flux, and nothing maintained itself fixed for all time, there would no longer be any possibility of getting to know about the world, and everything would be plunged into confusion.”

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Gottlob Frege

Translation J. L. Austin (Oxford, 1950) as quoted by Stephen Toulmin, Human Understanding: The Collective Use and Evolution of Concepts (1972) Vol. 1, p. 55.
Grundgesetze der Arithmetik, 1893 and 1903

“Since it is only in the context of a proposition that words have any meaning, our problem becomes this: To define the sense of a proposition in which a number-word occurs.”

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Gottlob Frege

Nur im Zusammenhange eines Satzes bedeuten die Wörter etwas. Es wird also darauf ankommen, den Sinn eines Satzes zu erklären, in dem ein Zahlwort vorkommt.
Gottlob Frege (1950 [1884]). p. 73

“Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician.”

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Gottlob Frege

Attributed to Frege in: A. A. B. Aspeitia (2000), Mathematics as grammar: 'Grammar' in Wittgenstein's philosophy of mathematics during the Middle Period, Indiana University, p. 25

“I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction.”

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Gottlob Frege libro Los fundamentos de la aritmética

Gottlob Frege (1950 [1884]). The Foundations of Arithmetic. p. 99.

“Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic.”

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Gottlob Frege

"Letter to Bertrand Russel" (1902) in J. van Heijenoort, ed., From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931 (1967)

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Gottlob Frege Sense and reference

The discovery that the rising sun is not new every morning, but always the same, was one of the most fertile astronomical discoveries. Even to-day the identification of a small planet or a comet is not always a matter of course. Now if we were to regard equality as a relation between that which the names 'a' and 'b' designate, it would seem that a = b could not differ from a = a (i.e. provided a = b is true). A relation would thereby be expressed of a thing to itself, and indeed one in which each thing stands to itself but to no other thing.
As cited in: M. Fitting, Richard L. Mendelsoh (1999), First-Order Modal Logic, p. 142. They called this Frege's Puzzle.
Über Sinn und Bedeutung, 1892

“I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori.”

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Gottlob Frege libro Los fundamentos de la aritmética

Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction.
Gottlob Frege (1950 [1884]). The Foundations of Arithmetic. p. 99.

“A philosopher who has no connection to geometry is only half a philosopher, and a mathematician who has no philosophical vein is only half a mathematician.”

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Gottlob Frege

Original: (de) Ein Philosoph, der keine Beziehung zur Geometrie hat, ist nur ein halber Philosoph, und ein Mathematiker, der keine philosophische Ader hat, ist nur ein halber Mathematiker.

Gottlob Frege: Erkenntnisquellen der Mathematik und der mathematischen Naturwissenschaften, 1924/1925, submitted to Wissenschaftliche Grundlagen; posthumously published in: Frege, Gottlob: Nachgelassene Schriften und Wissenschaftlicher Briefwechsel. Felix Meiner Verlag, 1990, p. 293