Showing papers on "Finitism published in 1999"
TL;DR: The connection of Hilbert's considerations to issues in the foundations of mathematics during the second half of the 19th century is sketched, the work that laid the basis of modern mathematical logic is described, and the first steps in the new subject of proof theory are analyzed.
Abstract: Hilbert's finitist program was not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration with Bernays during the period from 1917 to 1922. These notes reveal a dialectic progression from a critical logicism through a radical constructivism toward finitism; the progression has to be seen against the background of the stunning presentation of mathematical logic in the lectures given during the winter term 1917/18. In this paper, I sketch the connection of Hilbert's considerations to issues in the foundations of mathematics during the second half of the 19th century, describe the work that laid the basis of modern mathematical logic, and analyze the first steps in the new subject of proof theory. A revision of the standard view of Hilbert's and Bernays's contributions to the foundational discussion in our century has long been overdue. It is almost scandalous that their carefully worked out notes have not been used yet to understand more accurately the evolution of modern logic in general and of Hilbert's Program in particular. One conclusion will be obvious: the dogmatic formalist Hilbert is a figment of historical (de)construction! Indeed, the study and analysis of these lectures reveal a depth of mathematical-logical achievement and of philosophical reflection that is remarkable. In the course of my presentation many questions are raised and many more can be explored; thus, I hope this paper will stimulate interest for new historical and systematic work.
99 citations
TL;DR: The problem of rule finitism is malformed, deriving from misconceptions about the relation between understanding a rule and making an application of it as discussed by the authors, which raises the puzzle as to how a learner can possibly extend the rule to applications beyond those examples which have been given as instruction in the rule.
Abstract: Rule following is often made an unnecessary mystery in the philosophy of social science. One form of mystification is the issue of 'rule finitism', which raises the puzzle as to how a learner can possibly extend the rule to applications beyond those examples which have been given as instruction in the rule. Despite the claim that this problem originated in the work of Wittgenstein, it is clear that his philosophical method is designed to evaporate, not perpetuate, such problems. The supposed problem of rule finitism is malformed, deriving from misconceptions about the relation between understanding a rule and making an application of it.
18 citations
TL;DR: The authors interroge le statut de la Theorie des ensembles contemporaine a travers une analyse de ses differentes formes d'exposition, i.e.
Abstract: RESUME. — Cette etude interroge le statut de la Theorie des Ensembles contemporaine a travers une analyse de ses differentes formes d'exposition.
1 citations
01 Jan 1999
TL;DR: In this article, the authors suggest to apply Zeno's results which can be explored by some kind of interpretation, such as negative and positive dialectics, atomism, radical empiricism, finitism, infinitism, and indefinitism.
Abstract: Zeno’s Dichotomy aporia says: “Motion is impossible, because an object in motion must reach the half-way point before it gets to the end (Telos)”. In the recent philosophical literature there are several kinds of interpretations: negative and positive dialectics, atomism, radical empiricism, finitism, infinitism, indefinitism, etc. The scientific reflections on the paradoxes time to time produce different types of “resolutions” of these problems.[1] Most of these treatments use some kind of measure concept which can be questioned.[2] Instead of resolution, we suggest to apply Zeno’s results which can be explored by some kind of interpretation.
1 citations