Gottlob Frege: Basic Laws of Arithmetic, Volume 1
The German philosopher and mathematician Gottlob Frege (1848-1925) was the father of analytic philosophy and to all intents and purposes the inventor of modern logic. Basic Laws of Arithmetic, originally published in German in two volumes (1893, 1903), is Freges magnum opus. It was to be the pinnacle of Freges lifes work. It represents the final stage of his logicist project the idea that arithmetic and analysis are reducible to logic and contains his mature philosophy of mathematics and logic. The aim of Basic Laws of Arithmetic is to demonstrate the logical nature of mathematical theorems by providing gapless proofs in Frege's formal system using only basic laws of logic, logical inference, and explicit definitions. The work contains a philosophical foreword, an introduction to Frege's logic, a derivation of arithmetic from this logic, a critique of contemporary approaches to the real numbers, and the beginnings of a logicist treatment of real analysis. As is well-known, a letter received from Bertrand Russell shortly before the publication of the second volume made Frege realise that his basic law V, governing the identity of value-ranges, leads into inconsistency. Frege discusses a revision to basic law V written in response to Russells letter in an afterword to volume II. The continuing importance of Basic Laws of Arithmetic lies not only in its bearing on issues in the foundations of mathematics and logic but in its model of philosophical inquiry. Frege's ability to locate the essential questions, his integration of logical and philosophical analysis, and his rigorous approach to criticism and argument in general are vividly in evidence in this, his most ambitious work. Philip Ebert and Marcus Rossberg present the first full English translation of both volumes of Freges major work preserving the original formalism and pagination. The edition contains a foreword by Crispin Wright and an extensive appendix providing an introduction to Frege's formal system by Roy T. Cook.
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0S(nSRs 0SUR according Analysis argument places aS(aSMq Basic Law Begriffsschrift belongs calculating game Cantor cardinal number series chess concavity concept concept-script proposition Construction contentual arithmetic contraposition cS(oSp definition derive do(aoq domain of magnitudes dS(aSq equal equality-sign False figure-group ﬁrst first-level function follows formal arithmetic Frege function-name fundamental series German letter Gottlob Frege Grundgesetze horizontal IIIa IIIc inference iS(iSMq judgement-stroke logical modus ponens mS(dSRq mS(xSp notation nS(nSMq nS(ySp number-figures number-signs objects falling occur pLUbSs positival class proof proper name prove the proposition q-relation q-series qSs fs qSs pSs fs real numbers reference Relation named replace Roman letter rules rules of inference second-level function signs subcomponent supercomponent theory Thomae’s translation True truth-value v-concept value-range words x;mS(y;nSR(qPp xS(cSRq yS(ySMq Zero εΦ(ε