Abstract not found

We discuss the development of metamathematics in the Hilbert school, and Hilbert's proof-theoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.

century linguistics. In mid-century, he was a leading figure in American linguistics, serving as president of the Linguistic Society of America in 1955, just a year before Roman Jakobson. It is fair to say that during that decade—the years just before generative grammar came on the scene—Zellig Harris and Charles Hockett were the two

: A model of empirical knowledge and reasoning is introduced that can be used for tutoring and making expert diagnoses. The program - called Intellectual Hypertext (IH) - based on theories of logic empiricism and hypertext. For the reasoning IH uses fuzzy algorithm. Numeric and linguistic fuzzy marks are also supported. Keywords: Logic empirism, semantic model, fuzzy reasoning, hypertext It's a fact, if we asked ordinary people (any people in common life, not in science) what is the most difficult science for you, they say answeres theorems and so on. But if we asked mathematicians with the same question, they say - empirical knowledge, meaning, that it is very difficult to formalize it. For a long time philosophers try to create universal language of science [Russell, Whitehead 1910-1913]. And even make correspondence between rational numbers and judgments, text and probable function of science [Nalimov 1993]. They try to find criteria for judgment verification. But they didn't succ...

This article is essentially a part of my thesis for the degree DrSc (Doctor of Sciences). Therefore it mainly surveys my articles [41, 42, 43, 28, 29, 44, 22], and it is structured according to the requirements for such theses. I made only minor changes in the original text and added a few further references. Since Godel's main achievement concerns the problem of consistency and some of the problems that I am going to describe had been considered by him, I think that it is appropriate to publish this article in Godel Society. 1 Historical remarks The question that we are going to consider in is interesting per se and is related to some more practical questions, especially in complexity theory, but the original motivation for it comes from foundational studies. Among the variety of streams in foundations of mathematics, the one which had the biggest influence and which very much determined later development of mathematical logic was Hilbert's<F1

This article presents translations of works by Troyanskii describing his 1933 proposals for a mechanical translation device, some later elaborations of the linguistic and technical features, and also translated extracts from commentaries made in 1959 by the Russian editors of his papers.

This article was perhaps the single most influential publication in the early history of MT, convincing many outside the field that MT was a misguided activity, and it continues today to be cited as `evidence' of the impossibility of MT

This paper explores specific issues concerning linguistic theory and the use of simplicity criteria in the early Transformational Generative Grammar literature. In particular, the influence of Nelson Goodman and Willard Van Orman Quine upon the work of Noam Chomsky during the 1950s is assessed. The main topics considered include the development of constructional system theory, the use of mechanical procedures for measuring the formal simplicity of extralogical bases in constructional systems, and the way in which Chomsky adapted these techniques in order to facilitate the analysis of natural language. In this context, the influence of constructive nominalism upon Chomsky’s early work is also considered. Finally, the relationship between the notion of simplicity in 1950s-style generative grammar and more recent discussions of economy in the Minimalist Program is assessed.

Abstract In Carnap’s autobiography, he tells the story how one night in January 1931, “the whole theory of language structure ” in all its ramifications “came to [him] like a vision”. The shorthand manuscript he produced immediately thereafter, he says, “was the first version ” of Logical Syntax of Language. This document, which has never been examined since Carnap’s death, turns out not to resemble Logical Syntax at all, at least on the surface. Wherein, then, did the momentous insight of 21 January 1931 consist? We seek to answer this question by placing Carnap’s shorthand manuscript in the context of his previous efforts to accommodate scientific theories and metalinguistic claims within Wittgenstein’s Tractatus theory of meaning. The breakthrough of January 1931 consists, from this viewpoint, in the rejection of the Tractatus theory in favor of the meta-mathematical perspective of Hilbert, Gödel, and Tarski. This was not yet the standpoint of the published Logical Syntax, as we show, but led naturally to the “principle of tolerance ” and thus to Carnap’s mature philosophy, in which the inconsistencies between this first view and the principle of tolerance, which survived into the published Syntax, were overcome.

ABSTRACT: Here is a crude list, possibly summarizing the role of paradoxes within the framework of mathematical logic: 1. directly motivating important theories (e.g. type theory, axiomatic set theory, combinatory logic); 2. suggesting methods of proving fundamental metamathematical results (fixed point theorems, incompleteness, undecidability, undefinability); 3. applying inductive definability and generalized recursion; 4. introducing new semantical methods (e. g. revision theory, semi-inductive definitions, which require non-trivial set theoretic results); 5. (partly) enhancing new axioms in set theory: the case of anti-foundation AFA and the mathematics of circular phenomena; 6. suggesting the investigation of non-classical logical systems, from contraction-free and many-valued logics to systems with generalized quantifiers; 7. suggesting frameworks with flexible typing for the foundations of Mathematics and Computer Science; 8. applying forms of self-referential truth and in Artificial Intelligence, Theoretical Linguistics, etc. Below we attempt to shed some light on the genesis of the issues 1–8 through the history of the paradoxes in the twentieth century, with a special emphasis on semantical aspects.