Gotlob FREGE Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet, I-II [all published].

 

Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet, I-II [all published].

FREGE, Gotlob.

 

Détails bibliographiques

 

Titre : Grundgesetze der Arithmetik. ...

Éditeur : Hermann Pohle 1893-1903, Jena

Date d'édition : 1893

Edition : First edition

Description :

A fine copy of Frege’s magnum opus, from the library of American philosopher Paul Weiss. "The climax of Frege's career as a philospher should have been the publication of the two volumes of ‘Die Grundgesetze der Arithmetik’ (1893-1903), in which he set out to present in formal manner the logicist construction of arithmetic on the basis of pure logic and set theory. This work was to execute the task which had been sketched in the earlier books on the philosophy of mathematics: it was to enunciate a set of axioms which would be recognizably truths of logic, to propound a set of undoubtedly sound rules of inference, and then to present, one by one, derivations by these rules from these axioms of standard truths of arithmetic. The magnificent project aborted before it was ever completed. The first volume was published in 1893; the second volume did not appear until 1903 and while it was in the press Frege received a letter from Russell pointing out that the fifth of the initial axioms made the whole system inconsistent. This was the axiom which, in Frege's words, allowed 'the transition from a concept to its extension', the transition which was essential if it was to be established that numbers were logical objects. Frege’s system, with this axiom, permitted the formation of the class of all classes that are not members of themselves. But the formation of such a class, Russell pointed out, leads to paradox: if it is a member of itself then it is not a member of itself; if it is not a member of itself, then it is a member of itself. A system which leads to such paradox cannot be logically sound. With good reason, Frege was utterly downcast by this discovery, though he strove to patch his system by weakening the guilty axiom. We now know that his logicist programme cannot ever be successfully carried out. The path from the axioms of logic to the theorems of arithmetic is barred at two points. First, as Russell's paradox showed, the naive set theory which was part of Frege’s logical basis was inconsistent in itself, and the remedies which Frege proposed for this proved ineffective. Thus, the axioms of arithmetic cannot be derived from purely logical axioms in the way he hoped. Secondly, the notion of 'axioms of arithmetic' was itself latter called in question when Gödel showed that it was impossible to give arithmetic a complete and consistent axiomatization. None the less, the concepts and insights developed by Frege in the course of expounding his logicist thesis have a permanent interest which is unimpaired by the defeat of that programme at the hands of Russel and Gödel". (The Oxford Companion to Philosophy). Provenance: with the signature of Paul Weiss to the front free end paper. The American philosopher Paul Weiss (1901-2002) who taught at Bryn Mawr and Yale, co-edited (with Charles Hartshorne) the first six volumes of The Collected Papers of Charles Sanders Peirce (Harvard University Press, 1931-1935), and in 1947 founded the Metaphysical Society of America and its academic journal, Review of Metaphysics, serving as the journal’s editor until 1964. Two volumes. Large 8vo: 246 x 163 mm. The two volumes bound together in early twentieth century dark blue cloth with paper spine label, a fine and clean copy. XXXII, 253, (1); (2:blank),xv, (1), 265, (1) pp. N° de réf. du libraire 2309

 

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