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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 An investigation of earlier work by Brouwer (including an unpublished lecture in Geneva of 1934) in our opinion fully supports our position and shows that the method was not at all new for him.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
Concepts and Axioms
, 1998
"... The paper discusses the transition from informal concepts to mathematically precise notions; examples are given, and in some detail the case of lawless sequences, a concept of intuitionistic mathematics, is discussed. A final section comments on philosophical discussions concerning intuitionistic lo ..."
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The paper discusses the transition from informal concepts to mathematically precise notions; examples are given, and in some detail the case of lawless sequences, a concept of intuitionistic mathematics, is discussed. A final section comments on philosophical discussions concerning intuitionistic logic in connection with a "theory of meaning". What I have to tell here is not a new story, and it does not contain any really new ideas. The main difference with my earlier discussions of the same topics ([TD88, chapter16],[Tro91]) is in the emphasis. This paper starts with some examples of the transition from informal concepts to mathematically precise notions, followed by a more detailed discussion of one of these examples, the intuitionistic notion of a choice sequence, arguing for the lasting interest of this notion for the philosophy of mathematics. In a final section, I describe my own position relative to some of the philosophical discussions concerning intuitionistic logic in the wr...
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 ...
Choice Sequences: a Retrospect
, 1996
"... Introduction The topic of this talk will be the lasting interest of L.E.J. Brouwer's notion of choice sequence for the philosophy of mathematics. In the past here has been done a good deal of work on choice sequences, but in the last decade the subject is a bit out of fashion, for several reasons, ..."
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Introduction The topic of this talk will be the lasting interest of L.E.J. Brouwer's notion of choice sequence for the philosophy of mathematics. In the past here has been done a good deal of work on choice sequences, but in the last decade the subject is a bit out of fashion, for several reasons, which I shall not go into here. In this retrospective I want to take a look with you at a special aspect of choice sequences, namely their interest as an important "casestudy" in the philosophy of mathematics. How does mathematics arrive at its concepts, and discover the principles holding for those concepts? This is a typically philosophical question, more easily posed than answered. A procedure which certainly has played a role and still plays a role might be described as informally rigorous analysis of a concept That is to say,  given an informally described, but intuitively clear concept,  one analyzes the concept as carefully as possibl