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Brouwer’s incomplete objects
"... Abstract. The theory of the idealized mathematician has been developed to formalize a method that is characteristic for Brouwer’s papers after 1945. The method has been supposed to be radically new in his work. We replace the standard theory about this method by, we think, a more satisfactory one. W ..."
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Abstract. The theory of the idealized mathematician has been developed to formalize a method that is characteristic for Brouwer’s papers after 1945. The method has been supposed to be radically new in his work. We replace the standard theory about this method by, we think, a more satisfactory one. We do not use an idealized mathematician. We claim that it is the systematic application of incomplete sequences, already introduced by Brouwer in 1918, that makes the method special. An investigation of earlier work by Brouwer (including an unpublished lecture in Geneva of 1934) in our opinion fully supports our position and shows that the method was not at all new for him. Résumé. La théorie du mathématicien idéal a été développée pour formaliser une méthode caractéristique des travaux de Brouwer postérieurs à 1945. On a supposé que cette méthode représente une nouveauté importante. Nous en proposons une nouvelle théorie qui, croyonsnous, est plus adéquate que celle couramment acceptée. Nous n’y utilisons pas l’idée du mathématicien idéal, mais plutôt avanons que c’est l’application systématique des séquences incomplètes, déjà introduites par Brouwer en 1918, qui rend cette méthode particulire. Selon nous, un examen des travaux antérieurs de Brouwer (incluant les notes inédites d’un cours donné Genève en 1934) confirme notre thèse et montre que cette méthode n’était pas du tout nouvelle pour lui. 1
History of Constructivism in the 20th Century
"... notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providi ..."
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notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providing an x which satisfies A. Establishing :8xAx finitistically means: providing a particular x such that Ax is false. In this century, T. Skolem 4 was the first to contribute substantially to finitist 4 Thoralf Skolem 18871963 History of constructivism in the 20th century 3 mathematics; he showed that a fair part of arithmetic could be developed in a calculus without bound variables, and with induction over quantifierfree expressions only. Introduction of functions by primitive recursion is freely allowed (Skolem 1923). Skolem does not present his results in a formal context, nor does he try to delimit precisely the extent of finitist reasoning. Since the idea of finitist reasoning ...
Individual Choice Sequences in the Work of L.E.J. Brouwer
, 2002
"... Choice sequences are sequences not completely determined by a law. We state that the introduction of particular choice sequences by Brouwer in the late twenties was not recognised as such. We claim that their later use in the method of the creative subject was not traced back to this original use of ..."
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Choice sequences are sequences not completely determined by a law. We state that the introduction of particular choice sequences by Brouwer in the late twenties was not recognised as such. We claim that their later use in the method of the creative subject was not traced back to this original use of them and has been misinterpreted. We show where these particular choice sequences appear in the work of Brouwer and we show how they should be handled.