From ancient times to the present, the discovery and presentation of new proofs of previously established theorems has been a salient feature of mathematical practice. Why? What purposes are served by such endeavors? And how do mathematicians judge whether two proofs of the same theorem are essentially different? Consideration of such questions illuminates the roles that proofs play in the validation and communication of mathematical knowledge and raises issues that have yet to be resolved by mathematical logicians. The Appendix, in (...) which several proofs of the Fundamental Theorem of Arithmetic are compared, provides a miniature case study. (shrink)
As initially envisioned, Gödel's Collected Works were to include transcriptions of material from his mathematical workbooks. In the end that material, as well as some other manuscript items from Gödel's Nachlass, had to be left out. This note describes some of the unpublished items in the Nachlass that are likely to attract the notice of scholars and surveys the extent of shorthand transcription efforts undertaken hitherto. Some examples of sources outside Gödel's Nachlass that may be of interest to Gödel scholars (...) are also indicated. (shrink)
This article provides an English translation of a historic discussion on the foundations of mathematics, during which Kurt GÖdel first announced his incompleteness theorem to the mathematical world. The text of the discussion is preceded by brief background remarks and commentary.
Though regarded today as one of the most important results in logic, the compactness theorem was largely ignored until nearly two decades after its discovery. This paper describes the vicissitudes of its evolution and transformation during the period 1930-1970, with special attention to the roles of Kurt Gödel, A. I. Maltsev, Leon Henkin, Abraham Robinson, and Alfred Tarski.
According to several commentators, Kurt Godel's incompleteness discoveries were assimilated promptly and almost without objection by his contemporaries - - a circumstance remarkable enough to call for explanation. Careful examination reveals, however, that there were doubters and critics, as well as defenders and rival claimants to priority. In particular, the reactions of Carnap, Bernays, Zermelo, Post, Finsler, and Russell, among others, are considered in detail. Documentary sources include unpublished correspondence from Godel's Nachlass.
1. BACKGROUNDThe papers of Kurt Gödel were donated to the Institute for Advanced Study by his widow Adele shortly after his death in 1978. They were catalogued by the reviewer during the years 1982–1984 and subsequently placed on indefinite loan to the Rare Books and Manuscripts Division of the Firestone Library at Princeton University, where they were opened to scholars on 1 April, 1985. In 1982 work also commenced on the editing of Gödel’s Collected Works, and in 1998 a grant (...) from the Alfred P. Sloan Foundation enabled the preservation microfilming of all of Gödel’s Nachlass. Somewhat later copies of those microfilm reels, excluding Gödel’s correspondence, were made available for purchase through IDC Publishers in the Netherlands. Volume III of the Collected Works, devoted to a selection of Gödel’s unpublished essays and lectures, was originally intended to include extracts from his scientific notebooks as well. As cataloging of the Nachlass progressed, however, it became clear that the sheer extent of the material in those notebooks, much of it couched in the long-obsolete Gabelsberger shorthand, precluded carrying out that intention. In addition, exhaustion of funding, the editors’ flagging energies after twenty years of effort, and the need to train more scholars to read the Gabelsberger script demanded that the Collected Works project come to an end in 2003. In the hope, however, that others might eventually extend that work, the article ‘Future tasks for Gödel scholars’ [Dawson and Dawson, 2005], published two years later in the Bulletin of Symbolic Logic, surveyed the most important materials left unpublished. Now, happily, a new generation of scholars has begun to transcribe and edit some of the items listed there: Gödel’s logic lectures at Notre Dame [Adžić and Došen, 2017], his series of philosophical notebooks [Engelen, 2019–], and the present volume, one of several now in print or in preparation by Jan von Plato’s team at the University of Helsinki. (shrink)