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Logic

1956

Logic, Semantics, Metamathematics

philosophy

logic

math

mathematics

Alfred Tarski - 1956

Published with the aid of a grant from the National Endowment for the Humanities.Contains the only complete English-language text of The Concept of Truth in Formalized Languages.

Tarski made extensive corrections and revisions of the original translations for this edition, along with new historical remarks. It includes a new preface and a new analytical index for use by philosophers and linguists as well as by historians of mathematics and philosophy.

Remarks on the Foundations of Mathematics

Ludwig Wittgenstein - 1956

Wittgenstein's work remains, undeniably, now, that of one of those few philosophers who will be read by all future generations.The Remarks analyzes in depth such topics as logical compulsion (the “must”) and mathematical conviction; calculation as experiment; mathematical surprise, discovery, and invention; Russell's logic, Gödel's theorem, Cantor's diagonal procedure, Dedekind's cuts; the nature of proof and contradiction; and the role of mathematical propositions in the forming of concepts.Wittgenstein's later philosophy was much involved with the concept of “language-games,” of which mathematics was one.

It was his feeling that a proper analysis of the use of language would clarify concepts and lead to the solution of (what seem to be) philosophical problems.Sometimes, Wittgenstein's expository method is pre-Socratic: a flow of disconnected statements, not unlike Heraclitean fragments, that range from clear aphorisms to cryptic oracles. Elsewhere, there are brief Socratic dialogues with imaginary persons, opponents of equally severe seriousness, representatives of the other half of Wittgenstein strove for total clarity of language as a means of solving philosophical problems, but some of his most meaningful statements here are expressed suggestively, subjectively, poetically.

Introduction to Mathematical Logic (Pms-13), Volume 13

Alonzo Church - 1956

Logic is sometimes called the foundation of mathematics: the logician studies the kinds of reasoning used in the individual steps of a proof.

Alonzo Church was a pioneer in the field of mathematical logic, whose contributions to number theory and the theories of algorithms and computability laid the theoretical foundations of computer science. His first Princeton book, The Calculi of Lambda-Conversion (1941), established an invaluable tool that computer scientists still use today.Even beyond the accomplishment of that book, however, his second Princeton book, Introduction to Mathematical Logic, defined its subject for a generation. Originally published in Princeton's Annals of Mathematics Studies series, this book was revised in 1956 and reprinted a third time, in 1996, in the Princeton Landmarks in Mathematics series. Although new results in mathematical logic have been developed and other textbooks have been published, it remains, sixty years later, a basic source for understanding formal logic.Church was one of the principal founders of the Association for Symbolic Logic; he founded the Journal of Symbolic Logic in 1936 and remained an editor until 1979 At his death in 1995, Church was still regarded as the greatest mathematical logician in the world.