„If we wish to express our ideas in terms of the concepts synthetic and analytic, we would have to point out that these concepts are applicable only to sentences that can be either true of false, and not to definitions. The mathematical axioms are therefore neither synthetic nor analytic, but definitions. …Hence the question of whether axioms are a priori becomes pointless since they are arbitrary.“

The Philosophy of Space and Time (1928, tr. 1957)

Hans Reichenbach photo
Hans Reichenbach
professor académico alemão 1891 - 1953

Citações relacionadas

Eduard Jan Dijksterhuis photo
Walter A. Shewhart photo
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Georg Cantor photo

„Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established.“

—  Georg Cantor mathematician, inventor of set theory 1845 - 1918

From Kant to Hilbert (1996)
Contexto: Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real.

Hermann Grassmann photo

„From the imputation of confounding axioms with assumed concepts Euclid himself, however, is free. Euclid incorporated the former among his postulates while he separated the latter as common concepts—a proceeding which even on the part of his commentators was no longer understood, and likewise with modern mathematicians, unfortunately for science, has met with little imitation. As a matter of fact, the abstract methods of mathematical science know no axioms at all.“

—  Hermann Grassmann German polymath, linguist and mathematician 1809 - 1877

As quoted in "Diverse Topics: The Origin of Thought Forms," The Monist (1892) Vol. 2 https://books.google.com/books?id=8akLAAAAIAAJ&pg=PA120 ed., Paul Carus, citing The Open Court Vol. II. No. 77. A Flaw in the Foundation of Geometry by Hermann Grassmann, translated from his Ausdehnungslehre

Georg Cantor photo
James Martin (author) photo

„From a very early age, we form concepts. Each concept is a particular idea or understanding we have about our world. These concepts allow us to make sense of and reason about the things in our world. These things to which our concepts apply are called objects.“

—  James Martin (author) British information technology consultant and writer 1933 - 2013

James Martin (1993, p. 17) as cited in: " CIS330 Object Oriented Approach Ch2 http://webcadnet.blogspot.nl/2011/04/cis330-object-oriented-approach-text_3598.html" webcadnet.blogspot.nl. 2011/04/16

Leonhard Euler photo
Thomas Young (scientist) photo
Jacques Bainville photo

„Nothing is more false than the axiom that governments are belligerent and peoples are pacific.“

—  Jacques Bainville French historian and journalist 1879 - 1936

Action Française (3 July 1913), quoted in William R. Keylor, Jacques Bainville and the Renaissance of Royalist History in Twentieth-Century France (Baton Rouge: Louisiana State University Press, 1979), p. 65.

Immanuel Kant photo
Paul Bernays photo
Karl Mannheim photo

„In general there are two distinct and separable meanings of the term "ideology" — the particular and the total.
The particular conception of ideology is implied when the term denotes that we are sceptical of the ideas and representations advanced by our opponent. They are regarded as more or less conscious disguises of the real nature of a situation, the true recognition of which would not be in accord with his interests.“

—  Karl Mannheim Hungarian sociologist 1893 - 1947

Ideology and Utopia (1929)
Contexto: In general there are two distinct and separable meanings of the term "ideology" — the particular and the total.
The particular conception of ideology is implied when the term denotes that we are sceptical of the ideas and representations advanced by our opponent. They are regarded as more or less conscious disguises of the real nature of a situation, the true recognition of which would not be in accord with his interests. These distortions range all the way from conscious lies to half-conscious and unwitting disguises; from calculated attempts to dupe others to self-deception. This conception of ideology, which has only gradually become differentiated from the common-sense notion of the lie is particular in several senses. Its particularity becomes evident when it is contrasted with the more inclusive total conception of ideology. Here we refer to the ideology of an age or of a concrete historico-social group, e. g. of a class, when we are concerned with the characteristics and composition of the total structure of the mind of this epoch or of this group. Although they have something in common, there are also significant differences between them.

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Gottlob Frege photo

„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.“

—  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

Max Planck photo
Fernando Pessoa photo

„We never love someone. We just love the idea we have of someone. It's a concept of ours - summing up, ourselves - that we love.“

—  Fernando Pessoa, livro Livro do Desassossego

Ibid., p. 125
Original: Nunca amamos niguém. Amamos, tão-somente, a ideia que fazemos de alguém. É a um conceito nosso — em suma, é a nós mesmos — que amamos.
Fonte: The Book of Disquiet

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