Grundgesetze der Arithmetik. Begriffsgeschichtlich abgeleitet, 1893-1903 (first edition)
By FREGE, Gottlob
First edition, rare, complete in two volumes, of Frege s magnum opus, the aim of which was to provide rigorous, gapless proofs that arithmetic was entirely reducible to pure logic. In an attempt to realize Leibniz's ideas for a language of thought and a rational calculus, Frege developed a formal notation for regimenting thought and reasoning. Though this notation was first outlined in his Begriffsschrift (1879), the most mature statement of Frege's system was in his 2-volume Grundgesetze der Arithmetik (1893/1903) (Stanford Encyclopedia of Philosophy). The first volume of Grundgesetze focused on natural-number arithmetic, the second volume is concerned especially with the theory of real numbers, but a good deal of the space is taken up by criticism of current views, and at the end Frege allows that there is more to be done before real numbers can be properly defined. In a Nachwort, or appendix, written after most of the work had been printed, he sadly admits the ruin of his work by Bertrand Russell s discovery of a contradiction implicit in his premises; but he still maintains that his general conception of the relation between arithmetic and logic is correct, and considers methods by which the damage might be repaired (Kneale & Kneale, The Development of Logic (1962), p. 437). The projected third volume of Grundgesetze, which was to contain the definition of the real (and perhaps complex) numbers, and the derivation of the axioms of real (and perhaps complex) analysis, never appeared. The importance of Frege s ideas within contemporary philosophy would be hard to exaggerate. He was, to all intents and purposes, the inventor of mathematical logic, and the influence exerted on modern philosophy of language and logic, and indeed on general epistemology, by the philosophical framework within which his technical contributions were conceived and developed has been so deep that he has a strong case to be regarded as the inventor of much of the agenda of modern analytical philosophy itself. Grundgesetze was to have been the summit of Frege s life s work a rigorous demonstration within the system of Begriffsschrift how the fundamental laws of the classical pure mathematics of the natural and real numbers could be derived from principles which, in Frege s view, were purely logical. As is familiar, a letter received from Bertrand Russell shortly before the publication of the second volume made Frege realise that Axiom V of his system, governing identity for value-ranges, led to contradiction. Despite this contradiction, developments initiated in work by the late George Boolos and by Crispin Wright in the 1980s, showed that much of the main thrust of Frege s project can be salvaged. The broad upshot of this work was that the replacement of Axiom V in the system of Grundgesetze by what came to be known as Hume s Principle, associating one-one corresponding concepts with the same cardinal number, provided for, first, a consistent theory in which classical number theory could be developed and, second, a theory of considerable philosophical interest (MacTutor). Poggendorff IV, 1707; DSB V, 155. Two vols., large 8vo, pp. XXXII, 253, [1]; XV, [1], 265, [1]. Contemporary quarter-cloth and marbled boards, printed paper spine labels (that on Vol. 1 mostly rubbed away), binder's stamp on rear paste-down. Old library release stamp ('ausgeschieden') on half-title, title and first and last pages of text and cancelled library bookplate on front pastedown. Generally very good.
Published by: Jena: Hermann Pohle, 1893
Vendor: Landmarks of Science Books
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By FREGE, Gottlob
First edition, rare, complete in two volumes, of Frege s magnum opus, the aim of which was to provide rigorous, gapless proofs that arithmetic was entirely reducible to pure logic. In an attempt to realize Leibniz's ideas for a language of thought and a rational calculus, Frege developed a formal notation for regimenting thought and reasoning. Though this notation was first outlined in his Begriffsschrift (1879), the most mature statement of Frege's system was in his 2-volume Grundgesetze der Arithmetik (1893/1903) (Stanford Encyclopedia of Philosophy). The first volume of Grundgesetze focused on natural-number arithmetic, the second volume is concerned especially with the theory of real numbers, but a good deal of the space is taken up by criticism of current views, and at the end Frege allows that there is more to be done before real numbers can be properly defined. In a Nachwort, or appendix, written after most of the work had been printed, he sadly admits the ruin of his work by Bertrand Russell s discovery of a contradiction implicit in his premises; but he still maintains that his general conception of the relation between arithmetic and logic is correct, and considers methods by which the damage might be repaired (Kneale & Kneale, The Development of Logic (1962), p. 437). The projected third volume of Grundgesetze, which was to contain the definition of the real (and perhaps complex) numbers, and the derivation of the axioms of real (and perhaps complex) analysis, never appeared. The importance of Frege s ideas within contemporary philosophy would be hard to exaggerate. He was, to all intents and purposes, the inventor of mathematical logic, and the influence exerted on modern philosophy of language and logic, and indeed on general epistemology, by the philosophical framework within which his technical contributions were conceived and developed has been so deep that he has a strong case to be regarded as the inventor of much of the agenda of modern analytical philosophy itself. Grundgesetze was to have been the summit of Frege s life s work a rigorous demonstration within the system of Begriffsschrift how the fundamental laws of the classical pure mathematics of the natural and real numbers could be derived from principles which, in Frege s view, were purely logical. As is familiar, a letter received from Bertrand Russell shortly before the publication of the second volume made Frege realise that Axiom V of his system, governing identity for value-ranges, led to contradiction. Despite this contradiction, developments initiated in work by the late George Boolos and by Crispin Wright in the 1980s, showed that much of the main thrust of Frege s project can be salvaged. The broad upshot of this work was that the replacement of Axiom V in the system of Grundgesetze by what came to be known as Hume s Principle, associating one-one corresponding concepts with the same cardinal number, provided for, first, a consistent theory in which classical number theory could be developed and, second, a theory of considerable philosophical interest (MacTutor). Poggendorff IV, 1707; DSB V, 155. Two vols., large 8vo, pp. XXXII, 253, [1]; XV, [1], 265, [1]. Contemporary quarter-cloth and marbled boards, printed paper spine labels (that on Vol. 1 mostly rubbed away), binder's stamp on rear paste-down. Old library release stamp ('ausgeschieden') on half-title, title and first and last pages of text and cancelled library bookplate on front pastedown. Generally very good.
Published by: Jena: Hermann Pohle, 1893
Vendor: Landmarks of Science Books
